Full text of "Elementary Mathematics"

See other formats

00 a:

OU 160247 >m

OSMANIA UNIVERSITY LEBRAT

bode should be returicd on or before the date last mar

FELIX KLEIN

ELEMENTARY MATHEMATICS

FROM AN ADVANCED STANDPOINT

ARITHMETIC ALGEBRA - ANALYSIS

TRANSLATED FROM THE THIRD GERMAN SSlTION BY
E. R. HEDRICK AND C, A. NOBLE

PROFESSOR OF MATHEMATICS PROFESSOR OF MATHEMATICS

IN THE UNIVERSITY OF CALIFORNIA IN THE UNIVERSITY OF CALIFORNIA

AT LOS ANGELES AT BERKELEY

WITH 125 FIGURES

MACMILLAN AND CO., LIMITED

ST. MARTIN'S STREET, LONDON

1932

PRINTED IN GERMANY BY THE
SPAMERSCHE BUCHDRUCKEREI LEIPZIG

Preface to the First Edition.

The new volume which I herewith offer to the mathematical public,
and especially to the teachers of mathematics in our secondary schools,
is to be looked upon as a first continuation of the lectures Uber den
mathematischen Unterricht an den hoheren Schulen*, in particular, of those
on Die Organisation des mathematischen Unterrichts** ]by Schimmack and
me, which were published last year by Teubner. At that time our concern
was with the different ways in which the problem of instruction can be
presented to the mathematician. At present my concern is with deve-
lopments in the subject matter of instruction. I shall endeavor to put
before the teacher, as well as the maturing student, from the view-point
of modern science, but in a manner as simple, stimulating, and con-
vincing as possible, both the content and the foundations of the topics
of instruction, with due regard for the current methods of teaching.
I shall not follow a systematically ordered presentation, as do, for
example, Weber and Wellstein, but I shall allow myself free excursions
as the changing stimulus of surroundings may lead me to do in the
course of the actual lectures.

The program thus indicated, which for the present is to be carried
out only for the fields of Arithmetic, Algebra, and Analysis, was indicated
in the preface to Klein-Schimmack (April 1907). I had hoped then that
Mr.. Schimmack, in spite of many obstacles, would still find the time to
put my lectures into form suitable for printing. But I myself, in a way,
prevented his doing this by continuously claiming his time for work in
another direction upon pedagogical questions that interested us both.
It soon became clear that the original plan could not be carried out,
particularly if the work was to be finished in a short time, which seemed
desirable if it was to have any real influence upon those problems of
instruction which are just now in the foreground, As in previous years,
then, I had recourse to the more convenient method of lithographing
my lectures, especially since my present assistant, Dr. Ernst Hellinger,
showed himself especially well qualified for this work. One should not
underestimate the service which Dr. Hellinger rendered. For it is a
far cry from the spoken word of the teacher, influenced as it is by
accidental conditions, to the subsequently polished and readable record.

* On the teaching of mathematics in the secondary schools.
** The organization of mathematical instruction.

In precision of statement and in uniformity of explanations, the lecturer
stops short of what we are accustomed to consider necessary for a printed
publication.

I hesitate to commit myself to still further publications on the
teaching of mathematics, at least for the field of geometry. I prefer to
close with the wish that the present lithographed volume may prove
useful by inducing many of the teachers of our higher schools to renewed
use of independent thought in determining the best way of presenting
the material of instruction. This book is designed solely as such a mental
spur, not as a detailed handbook. The preparation of the latter I leave
to those actively engaged in the schools. It is an error to assume, as
some appear to have done, that my activity has ever had any other
purpose. In particular, the Lehrplan der Unterrichtskommission der Ge-
sellschaft Deutscher Naturforscher und Arzte* (the so-called "Meraner"
Lehrplan} is not mine, but was prepared, merely with my cooperation,
by distinguished representatives of school mathematics.

Finally, with regard to the method of presentation in what follows,
it will suffice if I say that I have endeavored here, as always, to combine
geometric intuition with the precision of arithmetic formulas, and that
it has given me especial pleasure to follow the historical development
of the various theories in order to understand the striking differences
in methods of presentation which parallel each other in the instruction
of today.

Gottingen, June, 1908

Klein.

Preface to the Third Edition.

After the firm of Julius Springer had completed so creditably the
publication of my collected scientific works, it off erred, at the suggestion
of Professor Courant, to bring out in book form those of my lecture
courses which, from 1890 on, had appeared in lithographed form and
which were out of print except for a small reserve stock.

These volumes, whose distribution had been taken over by Teubner,
during the last decades were, in the main, the manuscript notes of my
various assistants. It was clear to me, at the outset, that I could not
undertake a new revision of them without again seeking the help of
younger men. In fact I long ago expressed the belief that, beyond a
certain age, one ought not to publish independently. One is still
qualified, perhaps, to direct in general the preparation of an edition, but
is not able to put the details into the proper order and to take into proper
account recent advances in the literature. Consequently I accepted the

* Curriculum prepared by the commission on instruction of the Society of
German Natural Scientists and Physicians.

offer of Springer only after I was assured that liberal help in this respect
would be provided.

These lithographed volumes of lectures fall into two series. The
older ones are of special lectures which I gave from time to time, and
were prepared solely in order that the students of the following semester
might have at hand the material which I had already treated and .upon
which I proposed to base further work. These are the volumes on Non-
Euclidean Geometry, Higher Geometry, Hyper geometric Functions, Linear
Differential Equations, Riemann Surfaces, and Number Theory. In con-
trast to these, I have published several lithographed volumes of lectures
which were intended, from the first, for a larger circle of readers. These
are:

a) The volume on Applications of Differential and Integral Calculus
to Geometry, which was worked up from his manuscript notes by
C. H. Miiller. This was designed to bridge the gap between the needs
of applied mathematics and the more recent investigations of pure
mathematicians.

b) and c) Two volumes on Elementary Mathematics from an Advanced
Standpoint, prepared from his manuscript notes by E. Hellinger. These
two were to bring to the attention of secondary school teachers of mathe-
matics and science the significance for their professional work of their
academic studies, especially their studios in pure mathematics.

A thoroughgoing revision of the volumes of the second series seemed
unnecessary. A smoothing out, in places, together with the addition of
supplementary notes, was thought sufficient. With their publication
therefore, the initial step is taken. Volumes b), c), a) (in this order) will
appear as Parts I, II, III of a single publication bearing the title Ele-
mentary Mathematics from an Advanced Standpoint. The combining, in
this way, of volume a) with volumes b) and c) will meet the approval
of all who appreciate the growing significances of applied mathematics
for modern school instruction.

Meantime the revision of the volumes of the first series has begun,
starting with the volume on Non-Euclidean Geometry. But a more
drastic recasting of the material will be necessary here if the book is
to be a well-rounded presentation, and is to take account of the recent
advances of science. So much as to the general plan. Now a few words
as to the first part of the Elementary Mathematics.

I have reprinted the preface to the 1908 edition of b) because it
shows most clearly how the volume came into existence 1 . The second
edition (1911), also lithographed, contained no essential changes, and
the minor notes which were appended to it are now incorporated into

1 My co-worker, R. Schimmack, who is mentioned there, died in 1912 at the
age of thirty-one years, from a heart attack with which he was seized suddenly,
as he sat at his desk.

the text without special mention. The present edition retains 1 , in the
main, the text of the first edition, including such peculiarities as were
incident to the time of its origin. Otherwise it would have been necessary
to change the entire articulation, with a loss of homogeneity. But during
the sixteen years which have elapsed since the first publication, science
has advanced, and great changes have taken place in our school system,
changes which are still in progress. This fact is provided for in the
appendices which have been prepared, in collaboration with me, by
Dr. Seyfarth (Studienrat at the local Oberrealschule). Dr. Seyfarth also
made the necessary stylistic changes in the text, and has looked after
the printing, including the illustrations, so that I feel sincerely grateful
to him. My former co-workers, Messrs. Hellinger and Vermeil, as well
as Mr. A. Walther of Gottingen, have made many useful suggestions
during the proof reading. In particular, I am indebted to Messrs. Vermeil
and Billig for preparing the list of names and the index. The publisher,
Julius Springer has again given notable evidence of his readiness to
print mathematical works in the face of great difficulties.

Gottingen, Easter, 1Q24

Klein.

Preface to the English Edition.

Professor Felix Klein was a distinguished investigator. But he was
also an inspiring teacher. With the rareness of genius, he combined
familiarity with all the fields of mathematics and the ability to perceive
the mutual relations of these fields; and he made it his notable function,
as a teacher, to acquaint his students with mathematics, not as isolated
disciplines, but as an integrated living organism. He was profoundly
interested in the teaching of mathematics in the secondary schools, both
as to the material which should be taught, and as to the most fruitful
way in which it should be presented. It was his custom, during many
years, at the University of Gottingen, to give courses of lectures, prepared
in the interest of teachers and prospective teachers of mathematics in
German secondary schools. He endeavored to reduce the gap between
the school and the university, to rouse the schools from the lethargy
of tradition, to guide the school teaching into directions that would
stimulate healthy growth; and also to influence university attitude and
teaching toward a recognition of the normal function of the secondary
school, to the end that mathematical education should be a continuous
growth.

These lectures of Professor Klein took final form in three printed
volumes, entitled Elementary Mathematics from an Advanced Standpoint.

t1a/*A/1 in

VII

They constitute an invaluable work, serviceable alike to the university
teacher and to the teacher in the secondary school. There is, at present,
nothing else comparable with them, either with respect to their skilfully
integrated material, or to the fascinating way in which this material is
discussed. This English volume is a translation of Part I of the above
work. Its preparation is the result of a suggestion made by Professor
C our ant, of the University of Gottingen. It is the expression of a desire
to serve the need, in English speaking countries, of actual and prospective
teachers of mathematics; and it appears with the earnest hope that, in
a rather free translation, something of the spirit of the original has
been retained.

The Translators.

Contents

Page

Introduction 1

First Part: Arithmetic

I. Calculating with Natural Numbers 6

1. Introduction of Numbers in the Schools 6

2. The Fundamental Laws of Reckoning 8

3. The Logical Foundations of Operations with Integers 10

4. Practice in Calculating with Integers 17

II. The First Extension of the Notion of Number 22

1. Negative Numbers 23

2. Fractions 28

3- Irrational Numbers 31

III. Concerning Special Properties of Integers 37

IV. Complex Numbers 55

1. Ordinary Complex Numbers 55

2. Higher Complex Numbers, especially Quaternions 58

3. Quaternion Multiplication Rotation and Expansion 65

4. Complex Numbers in School Instruction 75

Concerning the Modern Development and the General Structure of

Mathematics 77

Second Part: Algebra

I. Real Equations with Real Unknowns 87

1. Equations with one parameter 87

2. Equations with two parameters 88

3. Equations with three parameters A, //, v 94

II. Equations in the field of complex quantities 101

A. The fundamental theorem of algebra 101

B. Equations with a complex parameter 104

1. The "pure" equation HO

2. The dihedral equation 115

3. The tetrahedral, the octahedral, and the icosahedral equations . 120

4. Continuation: Setting up the Normal Equation 124

5. Concerning the Solution of the Normal Equations 130

6. Uniformization of the Normal Irrationalities by Means of Trans-

cendental Functions 133

7. Solution in Terms of Radicals 138

8. Reduction of Genral Equations to Normal Equations 141

Contents. IX

Third Part: Analysis

J Page

I. Logarithmic and Exponential Functions 144

1. Systematic Account of Algebraic .Analysis 144

2. The Historical Development of the Theory 146

3. The Theory of Logarithms in the Schools 155

4. The Standpoint of Function Theory .156

II. The Goniometric Functions . . . 162

1. Theory of the Goniometric Functions 162

2. Trigonometric Tables 169

A. Purely Trigonometric Tables 170

B. Logarithmic Trigonometric Tables 172

3- Applications of Goniometric Functions 175

A. Trigonometry, in particular, spherical trigonometry 175

B. Theory of small oscillations, especially those of the pendulum . 186

C. Representation of periodic functions by means of series of gonio-
metric functions (trigonometric series) 190

III. Concerning Infinitesimal Calculus Proper 207

1. General Considerations in Infinitesimal Calculus 207

2. TAYLORS Theorem 223

3. Historical and Pedagogical Considerations 234

Supplement

I. Transcendence of the Numbers e and n 237

II. The Theory of Assemblages 250

1. The Power of an Assemblage 251

2. Arrangement of the Elements of an Assemblage 262

Index of Names 269

Index of Contents . 271

Introduction

In recent years 1 , a far reaching interest has arisen among university
teachers of mathematics and natural science directed toward a suitable
training of candidates for the higher teaching positions. This is really
quite a new phenomenon. For a long time prior to its appearance,
university men were concerned exclusively with their sciences, without
giving a thought to the needs of the schools, without even caring to
establish a connection with school mathematics. What was the result
of this practice? The young university student found himself, at the
outset, confronted with problems which did not suggest, in any particular,
the things with which he had been concerned at school. Naturally he
forgot these things quickly and thoroughly. When, after finishing his
course of study, he became a teacher, he suddenly found himself expected
to teach the traditional elementary mathematics in the old pedantic
way; and, since he was scarcely able, unaided, to discern any connection
between this task and his university mathematics, he soon fell in with
the time honored way of teaching, and his university studies remained
only a more or less pleasant memory which had no influence upon his
teaching.

There is now a movement to abolish this double discontinuity, helpful
neither to the school nor to the university. On the one hand, there is
an effort to impregnate the material which the schools teach with new
ideas derived from modern developments of science and in accord with
modern culture. We shall often have occasion to go into this. On the
other hand, the attempt is made to take into account, in university
instruction, the needs of the school teacher. And it is precisely in such
comprehensive lectures as I am about to deliver to you that I see one
of the most important ways of helping. I shall by no means address
myself to beginners, but I shall take for granted that you are all ac-
quainted with the main features of the chief fields of mathematics. I
shall often talk of problems of algebra, of number theory, of function
theory, etc., without being able to go into details. You must, therefore,
be moderately familiar with these fields, in order to follow me. My task
will always be to show you the mutual connection between problems in

f 1 Attention is again drawn to the fact that the wording of the text is, almost
throughout, that of the lithographed volume of 1908 and that comments which
refer to later years have been put into the appendices.]

Klein, Elementary Mathematics. 1

2 Introduction,

the various fields, a thing which is not brought out sufficiently in the
usual lecture cours.e, and more especially to emphasize the relation of
these problems to those of school mathematics. In this way I hope
to make it easier for you to acquire that ability which I look upon as
the real goal of your academic study: the ability to draw (in ample
measure) from the great body of knowledge there put before you a
living stimulus for your teaching.

Let me now put before you some documents of recent date which
give evidence of widespread interest in the training of teachers and
which contain valuable material for us. Above all I think here of the
addresses given at the last Meeting of Naturalists held September 16,
1907, in Dresden, to which body we submitted the Proposals for the
Scientific Training of Prospective Teachers of Mathematics and Science
of the Committee on Instruction of the Society of German Naturalists
and Physicians. You will find these Proposals as the last section in the
Complete Report of this Committee 1 which, since 1904, has been con-
sidering the entire complex of questions concerning instruction in mathe-
matics and natural science and has now ended its activity ; I urge you
to take notice, not only of these Proposals, but also of the other parts
of this very interesting report. Shortly after the Dresden meeting there
occurred a similar debate at the Meeting of German Philologists and
Schoolmen in Basel, September 25, in which, to be sure, the mathematical-
scientific reform movement was discussed only as a link in the chain
of parallel movements occurring in philological circles. After a report
by me concerning our aims in mathematical-natural science reform there
were addresses by P. Wendland (Breslau) on questions in Archeology,
Al. Brandl (Berlin) on modern languages and , finally , Ad. Harnack (Berlin)
on History and religion. These four addresses appeared together in one
broschure 2 to which I particulary refer you. I hope that this auspicious
beginning will develop into further cooperation between our scientists
and the philologists, since it will bring about friendly feeling and mutual
understanding between two groups whose relations have been unsympa-
thetic even if not hostile. Let us endeavor always to foster such good
relations even if we do among ourselves occasionally drop a critical
word about the philologists, just as they may about us. Bear in mind
that you will later be called upon in the schools to work together with
the philologists for the common good and that this requires mutual
understanding and appreciation.

Along with this evidence of efforts which reach beyond the borders
of our field, I should like to mention a few books which aim in the

1 Die Tdtigkeit der Unterrichtskommission der Gesellschajt deutscher Natur-
forscher und Arzte, edited by A. Gutzmer. Leipzig and Berlin, 1908.

2 Universitdt und Schule. Addresses delivered by F.Klein, P. Wendland,
Al. Brandl, Ad. Harnack. Leipzig 1907.

Introduction. 3

same direction in the mathematical field and which will therefore be
important for these lectures. Three years ago I gave, for the first time,
a course of lectures with a similar purpose. My assistant at that time,
R. Schimmack, worked the material up and the first part has recently
appeared in print 1 . In it are considered the different kinds of schools,
including the university, the conduct of mathematical instruction in
them, the interests that link them together, and other similar matters.
In what follows I shall from time to time refer to things which appear
there without repeating them. This makes it possible for me to extend
somewhat those considerations. That volume concerns itself with the
organization of school instruction. I shall now consider the mathematical
content of the material which enters into that instruction. If I frequently
advert to the actual conduct of instruction in the schools, my remarks
will be based not merely upon indefinite pictures of how the thing
might be done or even upon dim recollections of my own school days;
for I am constantly in touch with Schimmack, who is now teaching in
the Gottingen gymnasium and who keeps me informed as to the present
state of instruction, which has, in fact, advanced substantially beyond
what it was in earlier years. During this winter semester I shall discuss
"the three great AV, that is arithmetic, algebra, and analysis, with-
holding geometry for a continuation of the course during the coming
summer. Let me remind you that, in the language of the secondary
schools, these three subjects are classed together as arithmetic, and
that we shall often note deviations in the terminology of the schools as
compared with that at the universities. You see, from this small illustra-
tion, that only living contact can bring about understanding.

As a second reference I shall mention the three volume Enzyklopadie
der Elementarmathematik by H. Weber and J. Wellstein, the work which,
among recent publications, most nearly accords with my own tendencies.
For this semester, the first volume, Enzyklopadie der elementaren Algebra
und Analysis, prepared by H. Weber 2 , will be the most important. I
shall indicate at once certain striking differences between this work and
the plan of my lectures. In Weber- Wellstein, the entire structure of
elementary mathematics is built up systematically and logically in the
mature language of the advanced student. No account is taken of how
these things actually may come up in school instruction. The present-
ation in the schools, however, should be psychological and not syste-
matic. The teacher so to speak, must be a diplomat. He must take
account of the psychic processes in the boy in order to grip his interest ;

1 Klein, F., Vortrage uber den mathematischen Unterricht an hoheren Schulen.
Prepared by von R. Schimmack. Part 1. Von der Organisation des mathematischen
Unterrichts. Leipzig 1907. This book is referred to later as "Klein-Schimmack".

2 Second edition. Leipzig 1906. [Fourth edition, 1922, revised by P. Epstein.
Referred to as "Weber- Wellstein I".

4 Introduction.

and he will succeed only if he presents things in a form intuitively
comprehensible. A more abstract presentation will be possible only in
the upper classes. For example: The child cannot possibly understand
if numbers are explained axiomatically as abstract things devoid of
content, with which one can operate according to formal rules. On the
contrary, he associates numbers with concrete images. They are numbers
of nuts, apples, and other good things, and in the beginning they can
be and should be put before him only in such tangible form. While this
goes without saying, one should mutatis mutandis take it to heart,
that in all instruction, even in the university, mathematics should be
associated with everything that is seriously interesting to the pupil at
that particular stage of his development and that can in any way be
brought into relation with mathematics. It is just this which is back
of the recent efforts to give prominence to applied mathematics at the
university. This need has never been overlooked in the schools so much
as it has at the university. It is just this psychological value which I
shall try to emphasize especially in my lectures.

Another difference between Weber- Wellstein and myself has to do
with defining the content of school mathematics. Weber and Wellstein
are disposed to be conservative, while I am progressive. These things
are thoroughly discussed in Klein-Schimmack. We, who are called the
reformers, would put the function concept at the very center of in-
struction, because, of all the concepts of the mathematics of the past
two centuries, this one plays the leading role wherever mathematical
thought is used. We would introduce it into instruction as early as
possible with constant use of the graphical method, the representation
of functional relations in the xy system, which is used today as a matter
of course in every practical application of mathematics. In order to
make this innovation possible, we would abolish much of the traditional
material of instruction, material which may in itself be interesting, but
which is less essential from the standpoint of its significance in con-
nection with modern culture. Strong development of space perception,
above all, will always be a prime consideration. In its upper reaches,
however, instruction should press far enough into the elements of in-
finitesimal calculus for the natural scientist or insurance specialist to
get at school the tools which will be indispensable to him. As opposed
to these comparatively recent ideas, Weber- Wellstein adheres essentially
to the traditional limitations as to material. In these lectures I shall of
course be a protagonist of the new conception.

My third reference will be to a very stimulating book: Didaktik und
Methodik des Rechnens und der Mathematik 1 by Max Simon, who like

1 Second edition, Munich 1908. Separate reprint from Baumeister's Hand-
buck der Erziehungs- und Unterrichtslehre filr hohere Schulen, first edition, 1895.

Introduction. 5

Weber and Wellstein is at Strassburg. Simon is often in agrement
with our views, but he sometimes takes the opposite standpoint ; and
inasmuch as he is a very subjective, temperamental, personality he often
clothes these contrasting views in vivid words. To give one example,
the proposals of the Committee on Instruction of the Natural Scientists
require an hour of geometric propaedeutics in the second year of the
gymnasium, whereas at the present time this usually begins in the third
year. It has long been a matter of discussion which plan is the better;
and the custom in the schools has often changed. But Simon declares
the position taken by the Commission, which, mind you, is at worst
open to argument, to be ''worse than a crime", and that without in the
least substantiating his judgment. One could find many passages of
this sort. As a precursor of this book I might mention Simon's Methodik
der elementaren Arithmetik in Verbindung mit algebraischer Analysis 1 .
After this brief introduction let us go over to the subject proper,
which I shall consider under three headings, as above indicated.

Leipzig 1906.

First Part

Arithmetic

I. Calculating with Natural Numbers

We begin with the foundation of all arithmetic, calculation with
positive integers. Here, as always in the course of these lectures, we
first raise the question as to how these things are handled in the schools ;
then we shall proceed to the question as to what they imply when
viewed from an advanced standpoint.

1. Introduction of Numbers in the Schools

I shall confine myself to brief suggestions. These will enable you
to recall how you yourselves learned your numbers. In such an exposi-
tion it is, of course, not my purpose to induct you into the practice
of teaching, as is done in the Seminars of the secondary schools. I shall
merely exhibit the material upon which we shall base our critique.

The problem of teaching children the properties of integers and how
to reckon with them, and of leading them on to complete mastery, is
very difficult and requires the labor of several years, from the first school
year until the child is ten or eleven years old. The manner of instruction
as it is carried on in this field in Germany can perhaps best be designated
by the words intuitive and genetic, i. e., the entire structure is gradually
erected on the basis of familiar, concrete things, in marked contrast to
the customary logical and systematic method at the university.

One can divide up this material of instruction roughly as follows:
The entire first year is occupied with the integers from 1 to 20, the
first half being devoted to the range 1 to 10. The integers appear at
first as numbered pictures of points or as arrays of all sorts of objects
familiar to the children. Addition and multiplication are then presented
by intuitional methods, and are fixed in mind.

In the second stage, the integers from 1 to 100 are considered and the
Arabic numerals, together with the notion of positional value and the
decimal system, are introduced. Let us note, incidentally, that the name
"Arabic numerals", like so many others in science, is a misnomer.
This form of writing was invented by the Hindus, not by the Arabs.
Another principal aim of the second stage is knowledge of the multi-

Introduction of Numbers in the Schools. 7

plication table. One must know what 5 X 7 or 3 x 8 is in one's sleep,
so to speak. Consequently the pupil must learn the multiplication table
by heart to this degree of thoroughness, to be sure only after it has
been made clear to him visually with concrete things. To this end the
abacus is used to advantage. It consists, as you all know, of 10 wires
stretched one above another, upon each of which there are strung ten
movable beads. By sliding these beads in the proper way, one can
read off the result of multiplication and also its decimal form.

The third stage, finally, teaches calculation with numbers of more
than one digit, based on the known simple rules whose general validity
is evident, or should be evident, to the pupil. To be sure, this evidence
does not always enable the pupil to make the rules completely his own ;
they are often instilled with the authoritative dictum: "It is thus and
so, and if you don't know it yet, so much the worse for you!"

I should like to emphasize another point in this instruction which is
usually neglected in university teaching. It is that the application of
numbers to practical life is strongly emphasized. From the beginning,
the pupil is dealing with numbers taken from real situations, with coins,
measures, and weights; and the question, "What does it cost ?", which is so
important in daily life, forms the pivot of much of the material of instruc-
tion. This plan rises soon to the stage of problems, when deliberate
thought is necessary in order to determine what calculation is demanded.
It leads to the problems in proportion, alligation, etc. To the words
intuitive and genetic, which we used above to designate the character
of this instruction, we can add a third word, applications.

We might summarize the purpose of the number work by saying:
It aims at reliability in the use of the rules of operation, based on a parallel
development of the intellectual abilities involved, and without special concern
for logical relations.

Incidentally, I should like to direct your attention to a contrast
which often plays a mischievous role in the schools, viz., the contrast
between the university-trained teachers and those who have attended -
normal schools for the preparation of elementary school teachers. The
former displace the latter, as teachers of arithmetic, during or after
the sixth school year, with the result that a regrettable discontinuity
often manifests itself. The poor youngsters must suddenly make the
acquaintance of new expressions, whereas the old ones are forbidden.
A simple example is the different multiplication signs, the x being pre-
ferred by the elementary teacher, the point by the one who has attended
the university. Such conflicts can be dispelled, if the more highly
trained teacher will give more heed to his colleague and will try to meet
him on common ground. That will become easier for you, if you will
realize what high regard one must have for the performance of the
elementary school teachers. Imagine what methodical training is ne-

g Arithmetic: Calculating with Natural Numbers.

cessary to indoctrinate over and over again a hundred thousand stupid,
unprepared children with the principles of arithmetic ! Try it with your
university training; you will not have great success!

Returning, after this digression, to the material of instruction, we
note that after the third year of the gymnasium*, and especially in the
fourth year, arithmetic begins to take on the more aristocratic dress of
mathematics, for which the transition to operations with letters is charac-
teristic. One designates by a, b, c, or x, y 3 z any numbers, at first only
positive integers, and applies the rules and operations of arithmetic to
the numbers thus symbolized by letters, whereby the numbers are
devoid of concrete intuitive content. This represents such a long step
in abstraction that one may well declare that real mathematics begins
with operations with letters. Naturally this transition must not be
accomplished rapidly. The pupils must accustom themselves gradually
to such marked abstraction.

It seems unquestionably necessary that, for this instruction, the
teacher should know thoroughly the logical laws and foundations of
reckoning and of the theory of integers.

2. The Fundamental Laws of Reckoning

Addition and multiplication were familiar operations long before
any one inquired as to the fundamental laws governing these operations.
It was in the twenties and thirties of the last century that particularly
English and French mathematicians formulated the fundamental pro-
perties of the operations, but I will not enter into historical details here.
If you wish to study these, I recommend to you, as I shall often do,
the great Enzyklopddie der Mathematischen Wissenschaften mil Ein-
schlufi Hirer Anwendungen 1 , and also the French translation: Encyclope-
dic des Sciences mathematiques pures et appliquees 2 which bears in part
the character of a revised and enlarged edition. If a school library
has only one mathematical work, it ought to be this encyclo-
pedia, for through it the teacher of mathematics would be placed in
position to continue his work in any direction that might interest him.
For us, at this place, the article of interest is the first one in the first
volume 3 H.Schubert: "Grundlagen der Arithmetik", of which the trans-
lation into French is by Jules Tannery and Jules Molk.

* The German gymnasium is a nine-year secondary school, following a four-
year preparatory school. Hence the third year of the gymnasium is the student's
seventh school year.

1 Leipzig (B. G. Teubner) from 1908 on. Volume I has appeared complete,
Volumes II VI are nearing completion.

2 Paris (Gauthur-Villars) and Leipzig (Teubner) from 1904 on; unfortunately
the undertaking had to be abandoned after the death of its editor J. Molk (1914).

8 Arithmetik und Algebra, edited by W. Fr. Meyer (1896 1904) ; in the French
edition, the editor was J. Molk.

The Fundamental Laws of Reckoning. g

Going back to our theme, I shall enumerate the five fundamental
laws upon which addition depends:

1. a + b is always again a number, i. e., addition is always possible
(in contrast to subtraction, which is not always possible in the domain
of positive integers).

2. a + b is one-valued.

3. The associative law holds:

(a + b) + c = a+(b + c),

so that one may omit the parentheses entirely.

4. The commutative law holds:

a + b = b + a .

5. The monotonic law holds:

If b > c , then a + b > a + c .

These properties are all obvious immediately if one recalls the process
of counting; but they must be formally stated in order to justify logically
the later developments.

For multiplication there are five exactly analogous laws:

\. a b is always a number.

2. a b is one-valued.

3. Associative law: a (b c) = (a b) c = a b c.

4. Commutative law: a b = b a.

5. Monotonic law: If b > c, then a b > a c.

Multiplication together with addition obeys also the following law.

6. Distributive law:

a-(b + c)=a'b + a-c.

It is easy to show that all elementary reckoning can be based upon
these eleven laws. It will be sufficient to illustrate this fact by a simple
example, say the multiplication of 7 and 12. From the distributive
law we have:

7-12 = 7- (10 + 2) = 70 + 14,

and if we separate 14 into 10 + 4 (carrying the tens), we have, by the
associative law of addition,

70 + (10 + 4) = (70 + 10) + 4 = 80 + 4 = 84.

You will recognize in this procedure the steps of the usual decimal
reckoning. It would be well for you to construct for yourselves more
complicated examples. We might summarize by saying that ordinary
reckoning with integers consists in repeated use of the eleven fundamental
laws together with the memorized results of the addition and multiplication
tables.

But where does one use the monotonic laws? In ordinary formal
reckoning, to be sure, they are superfluous, but not in certain other

JO Arithmetic: Calculating with Natural Numbers.

problems. Let me remind you of the process called abridged multiplication
and division with decimal numbers 1 . That is a thing of great practical
importance which unfortunately is too little known in the schools, as
well as among university students, although it is sometimes mentioned
in the second year of the gymnasium. As an example, suppose that
one wished to compute 567 134, and that the units digit in each number
was of questionable accuracy, say as a result of physical measurement.
It would be unnecessary work, then, to determine the product exactly,
since one could not guarantee an exact result. It is, however, important
to know the order of magnitude of the product, i. e., to know between
which tens or between which hundreds the exact value lies. The mono-
tonic law supplies this estimate at once; for it follows by that law that
the desired value lies between 560-134 and 570 134 or between 560-130
and 570-140. I leave to you the carrying out of the details; at least
you see that the monotonic law is continually used in abridged reckoning.
A systematic exposition of these fundamental laws is, of course, not
to be thought of in the secondary schools. After the pupils have gained
a concrete understanding and a secure mastery of reckoning with
numbers, and are ready for the transition to operations with letters,
the teacher should take the opportunity to state, at least, the associative,
commutative, and distributive laws and to illustrate them by means
of numerous obvious numerical examples.

3. The Logical Foundations of Operations with Integers

While instruction in the schools will naturally not rise to still more
difficult questions, present mathematical investigation really begins with
the question : How does one justify the above-mentioned fundamental laws,
how does one account for the notion of number at all? I shall try to explain
this matter in accordance with the announced purpose of these lectures
to endeavor to get new light upon school topics by looking at them from
another point of view. I am all the more willing to do this because
these modern thoughts crowd in upon you from all sides during your
academic years, but not always accompanied by any indication of their
psychological significance.

First of all, so far as the notion of number is concerned, it is very
difficult to discover its origin. Perhaps one is happiest if one decides
to ignore these most difficult things. For more complete information
as to these questions, which are so earnestly discussed by the philo-
sophers, I must refer you to the article, already mentioned, in the
French encyclopedia, and I shall confine myself to a few remarks. A
widely accepted belief is that the notion of number is closely connected
with the notion of time, with temporal succession. The philosopher Kant

1 The monotonic laws will be used later, also, in the theory of irrational numbers.

The Logical Foundations of Operations with Integers. \\

and the mathematician Hamilton represent this view. Others think
that number has more to do with space perception. They base the
notion of number upon the simultaneous perception of different objects
which are near each other. Still others see, in number concepts, the
expression of a peculiar faculty of the mind which exists independently
of, and coordinate with, or even above, perception of space and time.
I think that this conception would be well characterized by quoting
from Faust the lines which Minkowski, in the preface of his book on
Diophantine Approximation, applies to numbers:

"Gottinnen thronen hehr in Einsamkeit,
Um sie kein Ort, noch weniger eine Zeit."

While this problem involves primarily questions of psychology and
epistemology, the justification of our eleven laws, at least the recent
researches regarding their compatibility, implies questions of logic. We
shall distinguish the following four points of view.

1. According to the first of these, best represented perhaps by Kant,
the rules of reckoning are immediate necessary results of perception,
whereby this word is to be understood, in its broadest sense, as "inner
perception 1 ' or intuition. It is not to be understood by this that mathe-
matics rests throughout upon experimentally controllable facts of ex-
ternal experience. To mention a simple example, the commutative law
is established by examining the accompanying picture, which
consists of two rows of three points each, that is, 2 3 = 3 2. If

the objection is raised that in the case of only moderately large
numbers, this immediate perception would not suffice, the reply is that
we call to our assistance the theorem of mathematical induction. If a
theorem holds for small numbers, and if an assumption of its validity for
a number n always insures its validity forn-{-\, then it holds generally
for every number. This theorem, which I consider to be really an in-
tuitive truth, carries us over the boundary where sense perception fails.
This standpoint is more or less that of Poincare in his well known
philosophical writings.

If we would realize the significance of this question as to the source
of the validity of our eleven fundamental rules of reckoning, we should
remember that, along with arithmetic, mathematics as a whole rests
ultimately upon them. Thus it is not asserting too much to say, that,
according to the conception of the rules of reckoning which we have
just outlined, the security of the entire structure of mathematics rests upon
intuition, where this word is to be understood in its most general sense.

2. The second point of view is a modification of the first. According
to it, one tries to separate the eleven fundamental laws into a larger
number of shorter steps of which one need take only the simplest
directly from intuition, while the remainder are deduced from these
by rules of logic without any further use of intuition. Whereas, before,

\2 Arithmetic: Calculating with Natural Numbers.

the possibility of logical operation began after the eleven fundamental
laws had been set up, it can start earlier here, after the simpler ones
have been selected. The boundary between intuition and logic is displaced
in favor of the latter. Hermann Grassmann did pioneer work in this
direction in his Lehrbuch der Arithmetik 1 in 1861. As an example from
it, I mention merely that the commutative law can be derived from
the associative law by the aid of the principle of mathematical induction.
Because of the precision of his presentation, one might place by the
side of this book of Grassmann one by the Italian Peano, Arithmetices
principia nova methodo exposita*. Do not assume, however, because of
this title, that the book was written in Latin ! It is written in a peculiar
symbolic language designed by the author to display each logical step
of the proof and emphasize it as such. Peano wishes to have a guarantee
in this way, that he is making use only of the principle which he specifi-
cally mentions, with nothing whatever coming from intuition. He wishes
to avoid the danger that countless uncontrollable associations of ideas
and reminders of perception might creep in if he used our ordinary
language. Note, too, that Peano is the leader of an extensive Italian
school which is trying in a similar way to separate into small groups
the premises of each individual branch of mathematics, and, with the
aid of such a symbolic language, to investigate their exact logical
connections.

3. We come now to a modern extension of these ideas, which has,
moreover, been influenced by Peano. I refer to that treatment of the
foundations of arithmetic which puts the theory of point sets into the
foreground. You will be able to form a notion of the wide range
of the idea of a point set if I tell you that the totality of all integers,
as well as that of all points on a line segment, are special examples
of point sets. Georg Cantor, as is generally known, was the first
to make this general idea the object of orderly mathematical
speculation. The theory of point sets, which he created, is now
claiming the profound attention of the younger generation of
mathematicians. Later I shall endeavor to give you a cursory view
of this subject. For the present, it is sufficient to characterize as follows
the tendency of the new foundation of arithmetic which have been based
upon it: The properties of integers and of operations with them are to
be deduced from the general properties and abstract relations of point sets,
in order that the foundation may be as sound and general as possible.

1 With the addition to the title "fur hohere Lehranstalten" (Berlin 1861).
The corresponding chapters are reprinted in H. Grassmann f s Gesammelten mathe-
matischen und physikalischen Werken (edited by F. Engel), Vol. II, 1, pp. 295 349-
Leipzig 1904.

2 Augustae Taurinorum. Torino 1889. [There is a more comprehensive
presentation in Peano's Formulaire de MatMmatiques (18921899).

The Logical Foundations of Operations with Integers. ja

One of the pioneers along this path was Richard Dedekind, who, in his
small but important book Was sind und was sollen die Zahlen? *, attempted
such a foundation for integers. H. Weber inclines to this point of view
in the first part of Weber-Wellstein, volume I (See p. 3). To be sure,
the deduction is quite abstract and offers, still, certain grave difficulties,
so that Weber, in an Appendix to Volume III 2 , gave a more elementary
presentation, using only finite point sets. In later editions, this appendix
is incorporated into Volume I. Those of you who are interested in such
questions are especially referred to this presentation.

4. Finally, I shall mention the purely formal theory of numbers, which,
indeed, goes back to Leibniz and which has recently been brought into
the foreground again by Hilbert. His address Vber die Grundlagen der
Logik und Arithmetik* at the Heidelberg Congress in 1904 is important
for arithemtic 3 . His fundamental conception is as follows: Once one
has the eleven fundamental rules of reckoning, one can operate with the
letters a, b, c t . . ., which actually represent arbitrary integers, without
bearing in mind that they have a real meaning as numbers. In other
words: let a, b, c, . . . , be things devoid of meaning, or things of whose
meaning we know nothing; let us agree only that one may combine
them according to those eleven rules, but that these combinations need
not have any real known meaning. Obviously one can than operate
with a,b,c, . . ., precisely as one ordinarily does with actual numbers.
Only the question arises here whether these operations could lead one to
contradictions. Now ordinarily one says that intuition shows us the
existence of numbers for which these eleven laws hold, and that it is
consequently impossible for contradictions to lurk in these laws. But
in the present case, where we are not thinking of the symbols as having
definite meaning, such an appeal to perception is not permissible. In
fact, there arises the entirely new problem, to prove logically that no oper-
ations with our symbols which are based on the eleven fundamental laws
can ever lead to a contradiction, i. e., that these eleven laws are consistent,
or compatible. While we were discussing the first point of view, we took
the position that the certainty of mathematics rests upon the existence
of intuitional things which fit its theorems. The adherents of this formal
standpoint, on the other hand, must hold that the certainty of mathematics
rests upon the possibility of showing that the fundamental laws considered
formally and without reference to their intuitional content, constitute a
.logically consistent system.

1 Braunschweig 1888; third edition 1911.

2 Angewandte Elementarmathematik. Revised by H. Weber, J. Wells tein,
R. H.Weber. Leipzig 1907-

* On the foundations of logic and arithmetic.

3 Verhandlungen des 3. international Mathematikerkongr esses in Heidelberg
August 813, 1904, p. 174 et seq., Leipzig 1905.

{4 Arithmetic: Calculating with Natural Numbers.

I shall close this discussion with the following remarks:

a) Hilbert indicated all of these points of view in his Heidel-
berg address, but he followed none of them through completely.
Afterwards he pushed them somewhat farther in a course of lectures,
but then abandoned them. We can thus say that here is a field for
investigation 1 .

b) The tendency to crowd intuition completely off the field and to
attain to really pure logical investigations seems to me not completely
feasible. It seems to me that one must retain something, albeit a minimum,
of intuition. One must always use a certain intuition in the most ab-
stract formulation with the symbols one uses in operations, in order to
recognize the symbols again, even if one thinks only about the shape of
the letters.

c) Let us even assume that the proposed problem has been solved
in a way free from objection, that the compatibility of the eleven funda-
mental laws has been proved logically. Precisely at this point an opening
is offered for a remark which I should like to make with the utmost
emphasis. One must see clearly that the real arithmetic, the theory of actual
integers, is neither established, nor can ever be established, by considerations
of this nature. It is impossible to show in a purely logical way that the
laws whose consistency is established in that manner are actually valid
for the numbers with which we are intuitionally familiar; that the
undefined things of which we speak, and the operations which we apply
to them, can be identified with actual numbers and with the processes
of addition and multiplication in their intuitively clear significance.
What is accomplished is, rather, that the tremendous problem of building
the foundations of arithmetic, unassailable in its complexity, is split into
two parts, and that the first, the purely logical problem, the setting up
of independent fundamental laws or axioms and the investigation of them
as to independence and consistency has been made available to study.
The second, the more epistemological part of the problem, which has
to do with the justification for the application of these laws to actual
conditions, is not even touched, although it must of course be solved
also if one will really build the foundations of arithmetic. This second
part presents, in itself, an extremely profound problem, whose diffi-
culties lie in the general field of epistemology. I can characterize its
standing most clearly perhaps, by the somewhat paradoxical remark
that anyone who tolerates only pure logic in investigations in pure
mathematics must, to be consistent, look upon the second part of the
problem of the foundation of arithmetic, and hence upon arithmetic
itself, as belonging to applied mathematics.

P Concerning more recent developments in these investigations, see the pre-
ceding footnote.]

The Logical Foundations of Operations with Integers. \ 5

I have felt obliged to go into detail here very carefully, in as much
as misunderstandings occur so often at this point, because people simply
overlook the existence of the second problem. This is by no means the
case with Hilbert himself, and neither my disagreement nor my agree-
ment with him is a warranted conclusion if it be based on such an
assumption.

Thomae of Jena, coined the neat expression "thoughtless thinkers"
for those persons who confine themselves exclusively to these abstract
investigations concerning things that are devoid of meaning, and to
theorems that tell nothing, and who forget not only that second problem
but often also all the rest of mathematics. This facetious term cannot
apply, of course, to people who carry on those investigations alongside
of many others of a different sort.

In connection with this brief survey of the foundation of arithmetic,
I shall bring to your notice a few general matters. Many have thought
that one could, or that one indeed must, teach all mathematics deduc-
tively throughout, by starting with a definite number of axioms and de-
ducing everything from these by means of logic. This method, which
some seek to maintain upon the authority of Euclid, certainly does not
correspond to the historical development of mathematics. In fact,
mathematics has grown like a tree, which does not start at its tiniest
rootlets and grow merely upward, but rather sends its roots deeper
and deeper at the same time and rate that its branches and leaves are
spreading upward. Just so if we may drop the figure of speech , mathe-
matics began its development from a certain standpoint corresponding
to normal human understanding, and has progressed, from that point,
according to the demands of science itself and of the then prevailing
interests, now in the one direction toward new knowledge, now in the
other through the study of fundamental principles. For example, our
standpoint today with regard to foundations is different from that of
the investigators of a few decades ago ; and what we today would state
as ultimate principles, will certainly be outstripped after a time, in
that the latest truths will be still more meticulously analyzed and
referred back to something still more general. We see, then, that as
regards the fundamental investigations in mathematics, there is no final
ending, and therefore, on the other hand, no first beginning, which could
offer an absolute basis for instruction.

Still another remark concerning the relation between the logical and
the intuitional handling of mathematics, between pure and applied
mathematics. I have already emphasized the fact that, in the schools,
applications accompany arithmetic from the beginning, that the pupil
learns not only to understand the rules, but to do something with them.
And it should always be so in the teaching of mathematics ! Of course,
the logical connections, one might say the rigid skeleton in the mathematical

16 Arithmetic: Calculating with Natural Numbers.

organism, must remain, in order to give it its peculiar trustworthiness.
But the living thing in mathematics, its most important stimulus, its
effectiveness in all directions, depends entirely upon the applications,
i. e., upon the mutual relations between those purely logical things and
all other domains. To banish applications from mathematics would be
comparable to seeking the essence of the living animal in the skeleton
alone, without considering muscles, nerves and tissues, instincts, in short,
the very life of the animal.

In scientific investigation there is often, to be sure, a division of labor
between pure and applied science, but when this happens, provision
must be made otherwise for maintaining their connection if conditions
are to remain sound. In any case, and this should be especially emphasiz-
ed here, for the school such a division of labor, such a fareaching specializ-
ation of the individual teacher, is not possible. To put the matter crassly,
imagine that at a certain school a teacher is appointed who treats
numbers only as meaningless symbols, a second teacher who knows how
to bridge the gap frdm these empty symbols to actual numbers, a third,
a fourth,. a fifth, finally, who understands the application of these
numbers to geometry, to mechanics, and to physics; and that these
different teachers are all turned lose upon the pupils. You see that
such an organization of teaching is impossible. In this way, the things
could not be brought to the comprehension of the pupils, neither would
the individual teachers be able even to understand each other. The
needs of school instruction itself require precisely a certain many sided-
ness of the individual teacher, a comprehensive orientation in the field
of pure and applied mathematics, in the broadest sense, and in-
clude thus a desirable remedy against a too extensive splitting up of
science.

In order to give a practical turn to the last remarks I refer again to
our above mentioned Dresden Proposals. There we recommend outright
that applied mathematics, which since 1898 has been a special subject
in the examination for prospective teachers, be made a required part
in all normal mathematical training, so that competence to teach pure
and applied mathematics should always be combined. In addition to
this, it should be noted that, in the Meran Curriculum 1 of the Commis-
sion of Instruction, the following three tasks are announced as the
purpose of mathematical instruction in the last school year:

\. A scientific survey of the systematic structure of mathematics.

2. A certain degree of skill in the complete handling, numerical and
graphical, of problems.

1 Reformvorschl&ge fur den mathematischen und naturwissenschaftlichen Unter-
richt, iiberreicht der Versammlung der Naturforscher und Arzte zu Meran. Leipzig,
1905. See also a reprint in the Gesamtbericht der Kommission, p. 93, as well as
in Klein-Schimmack, p. 208.

Practice in Calculating with Integers. \j

3. An appreciation of the significance of mathematical thought for a
knowledge of nature and for modern culture.

All these formulations I approve with deep conviction.

4. Practice in Calculating with Integers

Turning from discussions which have been chiefly abstract, let us
give our attention to more concrete things by considering the carrying
out of numerical calculation. As suitable literature for collateral reading,
I should mention first of all, the article on Numerisches Rechnen by
R. Mehnicke 1 in the Enzyclopadie. I can best give you a general view
of the things that belong here by giving a brief account of this article.
It is divided into two parts: A. Die Lehre vom genauen Rechnen*, and
B. Die Lehre vom gendherten Rechnen**. Under A occur all methods
for simplifying exact calculation with large integers. Convenient devices
for calculating, tables of products and squares, and in particular, calcu-
lating machines, which we shall discuss soon. Under B, on the other
hand, one finds a discussion of the methods and devices for all calculating
in which only the order of magnitude of the result is important, especially
logarithmic tables and allied devices, the slide rule, which is only an
expecially well-arranged graphical logarithmic table; finally, also, the
numerous important graphical methods. In addition to this reference I
can recommend the little book by J. Liiroth, Vorlesungen uber nume-
risches Rechnen****, which, written in agreeable form by a master of the
subject, gives a rapid survey of this field.

From the many topics that have to do with calculating with integers,
I shall select for discussion only the calculating machine, which you will
find in use, in a great variety of ingenious forms, by the larger banks
and business houses, and which is really of the greatest practical signi-
ficance. We have in our mathematical collection one of the most widely
used types, the "Brunsviga", manufactured by the firm Brunsviga-
Maschinenwerke Grimme, Natalis & Co. A.-G. in Braunschweig. The design
originated with the Swedish engineer Odhner, but it has been much chan-
ged and improved. Is hall describe the machine here in some detail, as a
typical example. You will find other kinds described in the books mentioned
above 3 . My description of course can give you a real understanding of the

1 Enzyklopadie der mathematischen Wissenschaften, Band I, Teil II. See
also v. Sanden, H., Practical Mathematical Analysis (Translation by Levy), Button
& Co. Horsburgh, E. M., Modern Instruments and Methods of Calculation.
Bell & Sons.

* The Theory of Exact Calculation.
** The Theory of Approximate Calculation.

2 Leipzig 1900.

*** Lectures on Numerical Calculation.

[ 3 Concerning other types of calculating machines, see also A. Galle, Mathe-
matische Instrumente, Leipzig 1912.]

Klein. Elementary Mathematics. 2

Arithmetic: Calculating with Natural Numbers.

machine only if you examine it afterwards personally and if you see,
by actual use, how it is operated. The machine will be at your disposal,
for that purpose, after the lecture.

So far as the external appearance of the Brunsviga is concerned, it
presents schematically a picture somewhat as follows (see Fig. 1, p. 18).
There is a fixed frame, the "drum", below which and sliding on it, is
a smaller longish case, the "slide". A handle which projects from the
drum on the right, ^is operated by hand. On the drum there is a series
of parallel slits, each of which carries the digits 0,1, 2,. ..,9, read
downwards; a peg s projects from each slit and can be set at pleasure
at any one of the ten digits. Corresponding to each of these slits there
is an opening on the slide under which a digit can appear. Figure 3, p. 19
gives a view of a newer model of the machine.

I think that the arrangement of the machine will be clearer if I
describe to you the process of carrying out a definite calculation, and

the way in which the machine
brings it about. For this I select
Multiplication.

The procedure is as follows:
One first sets the drum pegs on the
multiplicand, i. e., beginning at
the rigftt, one puts the first
lever at the one's digit, the se-
cond at the ten's digit of the
multiplicand, etc. If, for example,
the multiplicand is 12, one sets
the first lever at 2, the second
lever at 1 ; all the other levers
remain at zero (see Fig. 1).. Now
turn the handle once around,
clockwise. The multiplicand ap-
pears under the openings of the slide, in our case a 2 in the first opening
from the right, a 1 in the second, while zeros remain in all the others.
Simultaneously, however, in the first of a series of openings in the slide, at
the left, the digit 1 appears to indicate that we have turned the handle
once (Fig. 2). // now one has to do with a. multiplier of one digit, one turns
the handle as many times as this digit indicates; the multiplier will then
be exhibited on the slide to the left, while the product will appear on the
slide to the right. How does the apparatus bring this result about ? In
the first place there is attached to the under side of the slide, at the
left, a cogwheel which carries, equally spaced on its rim, the digits
0, 1 , 2, . . . , 9- By means of a driver, this cogwheel is rotated through
one tenth of its perimeter with every turn of the handle, so that a digit
becomes visible through the opening in the slide, which actually indicates

|i|

o
1

r-i

5
6

' '

)

Fig. i. Before the first turn.

2
3

I i

7
3
*

Fig. 2. After the first, turn.

Practice in Calculating with Integers.

Fig. 3.

the number of revolutions, in other words the multiplier. Now as to
the obtaining of the product, it is brought about by similar cogwheels,
one under each opening at the right of the slide. But how is it that
by one and the same turning of the handle, one of these wheels, in the
above case, moves by
one unit, the other
by two? This is
where the peculiarity
in construction of the
Brunsviga appears.
Under each slit of the
drum there is a flat
wheel-shaped disc
(driver) attached to
the axle of the handle,
upon which there are
nine teeth which are
movable in a radial di-
rection (see Fig. 4). By
means of the projecting
peg 5 , mentioned above,
one can turn a ringf/?'
which rests upon the
periphery of the disc,
so that, according to
the mark upon which
one sets S in the slit,
0, 1, 2, . . ., 9 of the
movable teeth spring outward (in
Fig. 4, two teeth). These teeth
engage the cogs under the corre-
sponding openings of the slide, so
that with one turn of the handle
each driver thrusts forward the corre-
sponding cogwheel by as many units
as there are teeth pushed out, i.e., by
as many teeth as one has set with the
corresponding peg S. Accordingly,
in the above illustration, when we start at the zero position, and
turn the handle once, the units wheel must jump to 2, the ten's
wheel to 1, so that 12 appears. A second turn of the handle moves
the units wheel another 2 and the tens wheel another 1, so that 24 ap-
pears, and similarly, we get, after 3 or 4 times, 3 12 = 36 or 4 12 = 48,
respectively.

Fig. 3 a.

Cogwheel

Driving wheel

Arithmetic: Calculating with Natural Numbers.

But now turn the handle a fifth time: Again, according to the
account above, the units wheel should jump again by two units, in other
words back to 0, the tens wheel by one, or to 5, and we should have
the false result 5 12 = 50. In the actual turning, however, the slide
shows 50, to be sure, until just before the completion of the turn; but
at the last instant the 5 changes into 6, so that the correct result appears.
Something has come into action now that we have not yet described,
and which is really the most remarkable point of such machines: the
so called carrying the tens. Its principle is as follows: when one of the
number bearing cogwheels under the slide (e. g., the units wheel) goes
through zero, it presses an otherwise inoperative tooth of the neighboring
driver (for the tens) into position, so that it engages the corresponding
cogwheel (the tens wheel) and pushes this forward one place farther than
it would have gone otherwise. You can understand the details of this
construction only by examining the apparatus itself. There is the less
need for my -going into particulars here because it is just the method
of carrying the tens that is worked out in the greatest variety of ways
in the different makes of machines, but I recommend a careful examina-
tion of our machine as an example of a most ingenious model. Our
collection contains separately the most important parts of the Brunsviga
which are for the .most part invisible in the assembled machine so
that you can, by examining them, get a complete picture of its ar-
rangement.

We can best characterize the operation of the machine, so far as
we have made its acquaintance, by the words adding machine, because,
with every turn of the handle, it adds, once, to the number on the slide at
the right, the number which has been set on the drum.

Finally, I shall describe in general that arrangement of the machine
which permits convenient operation with multipliers of more than one

digit. If we wish to calculate, say,
15 12 we should have to turn the
handle fifteen times, according to
the plan already outlined; moreover,
if one wished to have the multiplier
indicated by the counter at the left
of the slide, it would be necessary
to have, there also, a device for
carrying the tens. Both of these

difficulties are avoided by the following arrangement 1 . We first perform
the multiplication by five, so that 5 appears on the slide at the left
and 60 at the right (see Fig. 5). Now we push the slide one place to the

1
2

i !

2
3

5
6

(

Fig. 5.

1 In the newer models the cogwheel device for "carrying over" is likewise
very complete. .

Practice in Calculating with Integers. 21

right, so that, as shown in Fig. 5, its units cogwheel is cut out, its
tens cogwheel is moved under the units slit of the drum, its hundreds
cogwheel under, the tens slit, etc., while, at the left, this shift brings
it about that the tens cogwheel, instead of the units, is connected
with the driver which the handle carries. If we now turn the handle
once, 1 appears at the left, in ten's place, so that we read 15; at the

right, however, we do not get the addition | , .^ but | ^ or, in

other words, 60 + 120, since the 2 is ' 'carried over" to the tens wheel,
the 1 to the hundreds wheel. Thus we get correctly 15 12 = 180..
It is, as you see, the exact mechanical translation of the customary process
of written multiplication, in which one writes down under one another,
the products of the multiplicand by the successive digits of the
multiplier, each product moved to the left one place farther than the
preceding, and then adds. In just the same way one proceeds quite
generally when the multiplier has three or more digits, that is, after the
usual multiplication by the ones, one moves the slide 1,2, ... places to
the right and turns the handle in each place as many times as the digit
in the tens, hundreds, . . . place of the multiplier indicates.

Direct examination of the machine will disclose how one can perform
other calculations with it; the remark here will suffice that subtraction
and division are effected by turning the handle in the direction opposite
to that employed in addition.

Permit me to summarize by remarking that the theoretical principle
of the machine is quite elementary and represents merely a technical
realization of the rules which one always uses in numerical calculation.
That the machine really functions reliably, that all the parts engage
one another with unfailing certainty, so that there is no jamming, that
the wheels do not turn farther than is necessary, is, of course, the
remarkable accomplishment of the man who made the design, and the
mechanician who carried it out.

Let us consider for a moment the general significance of the fact that
there really are such calculating machines, which relieve the mathematician
of the purely mechanical work of numerical calculation, and which do
this work faster, and, to a higher degree free from error, than he himself
could do it, since the errors of human carelessness do not creep into
the machine. In the existence of such a machine we see an outright
confirmation that the rules of operation alone, and not the meaning of
the numbers themselves, are of importance in calculating; for it is only
these that the machine can follow; it is constructed to do just that;
it could not possibly have an intuitive appreciation of the meaning of
the numbers. We shall not, then, wish to consider it as accidental that
such a man as Leibniz, who as both an abstract thinker of first rank
and a man of the highest practical gifts, was, at the same tine, both

22 Arithmetic: The First Extension of the Notion of Number.

the father of purely formal mathematics and the inventor of a calcu-
lating machine. His machine is, to this day, one of the most prized
possessions of the Kastner Museum in Hannover. Although it is not
historically authenticated, still I like to assume that when Leibniz
invented the calculating machine, he not only followed a useful purpose,
but that he also wished to exhibit, clearly, the purely formal character
of mathematical calculation.

With the construction of the calculating machine Leibniz certainly
did not wish to minimize the value of mathematical thinking, and yet it
is just such conclusions which are now sometimes drawn from the
existence of the calculating machine. If the activity of a science can be
supplied by a machine, that science cannot amount to much, so it is
said; and hence it deserves a subordinate place. The answer to such
arguments, however, is that the mathematician, even when he is himself
operating with numbers and formulas, is by no means an inferior counter-
part of the errorless machine, ''thoughtless thinker" of Thomae; but
rather, he sets for himself his problems with definite, interesting, and
valuable ends in view, and carries them to solution in appropriate and
original manner. He turns over to the machine only certain operations
which recur frequently in the same way, and it is precisely the mathe-
maticianone must not forget this who invented the machine for his
own relief, and who, for his own intelligent ends, designates the tasks
which it shall perform.

Let me close this chapter with the wish that the calculating machine,
in view of its great importance, may become known in wider circles
than is now the case. Above all, every teacher of mathematics should
become familiar with it, and it ought to be possible to have it demon-
strated in secondary instruction.

II. The First Extension of the Notion of Number

With the last section we leave operations with integers, and shall
treat, in a new chapter, the extension of the number concept. In the
schools it is customary, in this field, to take in order the following steps :

1. Introduction of fractions and operations with fractions.

2. Treatment of negative numbers, in connection with the beginnings
of operations with letters.

3. More or less complete presentation of the notion of irrational numbers
by examples that arise upon different occasions, which leads, then, gra-
dually, to the notion of the continuum of real numbers.

It is a matter of indifference in which order we take up the first
two points. Let us discuss negative numbers before fractions.

Negative Numbers. 2}

1. Negative Numbers

Let us first note, as to terminology, that in the schools, one speaks
of positive and negative numbers, inclusively, as relative numbers in
distinction from the absolute (positive) numbers, whereas, in universities
this language is not common. Moreover, in the schools one speaks of
"algebraic numbers" 1 along with relative numbers, an expression which
we in universities employ, as you know, in quite another sense.

Now, as to the origin and introduction of negative numbers, I can
be brief in my reference to source material; these things are already
familiar to you, or you can at least easily make them so with the help
the references I shall give. You will find a complete treatment, for
example, in Weber- Wellstein ; also, in very readable form, in H. Burk-
hardt's Algebraischer Analysis 2 . This book, moreover, you might well
purchase, as it is of moderate size.

The creation of negative numbers is motivated, as you know, by
the demand that the operation of subtraction shall be possible in all cases.
If a < b then a b is meaningless in the domain of natural integers;
a number c = b a does exist, however, and we write

a b = c

which we call a negative number. This definition at once justifies the
representation of all integers by means of the scale of equidistant points

I 1 1 1 1 1 1 1 1

A +1 +2 +3 +4

on a straight line the "axis of abscissas" which extends in both directions
from an pjip 1 "". Gn^ A a y rtr msider this picture as a common possession
of all educated persons today, and one can, perhaps, assume that it
owes its general dissemination, chit^y, to the thermometer scale. The
commercial balance, with its reckonii _ 'n debits and credits, affords
likewise a graphic and familiar picture of negative numbers.

Let us, however, realize at once and emphatically how extr" ^Jiu " v
difficult in principle is the step, which is taken in school when negative
numbers are introduced. Where the pupil before was accustomed to
represent visually by concrete numbers of things the numbers, and,
later, the letters, with which he operated, as well as the results which
he obtained by his operations, he finds it now quite different. He has
to do with something new, the "negative numbers", which have, imme-
diately, nothing in common with his picture of numbers of things, but
he must operate with them as though they had, although the operations

1 See, e. g. Mehler, Hauptsatze der Elementarmathematik, Nineteenth edition,
p. 77, Berlin, 1895-

2 Leipzig 1903. [Third edition, revised by G. Faber, 1920.] See also Fine, H.,
The Number-System of Algebra treated Theoretically and Historically, Heath.

24 Arithmetic: The First Extension of the Notion of Number.

have graphically a meaning much less clear than the old ones. Here,
for the first time, we meet the transition from concrete to formal mathe-
matics. The complete mastery of this transition requires a high order
of ability in abstraction.

We shall now inquire in detail what happens to the operations of
calculation when negative numbers are introduced. The first thing to
notice is that addition and subtraction coalesce, substantially: The
addition of a positive number is the subtraction of the equal and opposite
negative number, In this connection, Max Simon makes the amusing
remark that, whereas negative numbers were created to make the
operation of subtraction possible without any exception, subtraction as
an independent operation ceased to exist by virtue of that creation.
For this new operation of addition (including subtraction) in the domain
of positive and negative numbers the five formal laws stated before
hold without change. These are, in brief (see p. 9 et seq.) :

1. Always possible.

2. Unique.

3. Associative law.

4. Commutative law. "

5. Monotonic law.

Notice, in connection with 5, that a < b means, now, that a lies to
the left of b in the geometric representation, so that we have, for
example 2 < 1 , 3 < 2.

The chief point in the multiplication of positive and negative numbers
is the rule of signs, that a - ( c) = ( c) a = (a c), and ( c) ( c')
= + (c c'). Especially the latter rule: ' 'Minus times minus gives plus"
is often a dangerous stumbling block. \K* shall return preaymrjy to the
inner significance of these rules; jn^ J *iow we shall combine them into
a statement defining multiplv^u&n f a series of positive and negative
numbers: The absolute val* e of a product is equal to the product of the
afarfute values of the.*-t rs >' ^ s sign is positive or negative according as
an even or an oaa number of factors is negative. With this convention,
multiplication in the domain of positive and negative numbers has again
the following properties:

1. Always possible.

2. Unique.

3. Associative.

4. Commutative.

5. Distributive with respect to addition.

There is a change only in the monotonic law; in its place one has
the following law:

6. If a > b then a - c ^ b c according as c 5^ 0.

Let us inquire, now, whether these laws, considered again purely
formally, are consistent. We must admit at once, however, that a purely

Negative Numbers. 25

logical proof of consistency is as yet much less possible here than it is
in the case of integers. Only a reduction is possible, in the sense that
the present laws are consistent if the laws for integers are consistent.
But until this has been completed by a logical consistency proof for
integers, one will have to hold that the consistency of our laws is based
solely on the fact that there are intuitive things, with intuitive relations,
which obey these laws. We noted above, as such, the series of integral
points on the axis of abscissas and we need only indicate what the rules
of operation signify there: The addition %' = x + #, where a is fixed,
assigns to each point x a second point x', so that the infinite straight
line is simply displaced along itself by an amount a , to the right or to
the left, according as a is positive or negative. In an analogous manner,
the multiplication x' = a % represents a similarity transformation of
the line into itself, a pure stretching for a > , a stretching together
with a reflexion in the origin for a < .

Permit me now to explain how, historically, all these things arose.
One must not think that the negative numbers are the invention of
some clever man who menufactured them, together with their con-
sistency perhaps, out of the geometric representation. Rather, during
a long period of development, the use of negative numbers forced itself,
so to speak, upon mathematicians. Only in the nineteenth century,
after men had been operating with them for centuries, was the con-
sideration of their consistency taken up.

Let me preface the history of negative numbers with the remark
that the ancient Greeks certainly had no negative numbers, so that
one cannot yield them the first place, in this case, as so many people
are otherwise prone to do. One must attribute this invention to the
Hindus, who also created our system of digits and in particular our zero.
In Europe, negative numbers came gradually into use at the time of
the Renaissance, just as the transition to operating with letters had
been completed. I must not omit to mention here that this completion
of operations with letters is said to have been accomplished by Vieta
in his book In Artem Analyticam Isagoge 1 .

From the present point of view, we have the so called parenthesis
rules for operations with positive numbers, which are, of course, con-
tained in our fundamental formulas, provided one includes the correpond-
ing laws for subtraction. But I should like to take them up somewhat
in detail, by means of two examples, in order, above all, to show the
possibility of extremely simple intuitive proofs for them, proofs which
need consist only of the representation and of the word "Look"!, as
was the custom with the ancient Hindus.

\. Given a > b and c > a, where a t b,c are positive. Then a b
is a positive number and is smaller than c , that is, c (a b) must

1 Tours 1591.

26 Arithmetic: The First Extension of the Notion of Number.

exist as a positive number. Let us represent the numbers on the axis
of abscissas and note that the segment between the points b and a has
the length a b. A glance at the representation shows that, if we
take away from c the segment a 6, the result is the same as though
we first took away the entire segment a and then restored the part &, i. e.,

(1) c (a b) = c a + b .

2. Given a >6 and c > d; then a b and c d are positive integers.
We wish to examine the product (a b) (c d) ; for that purpose
1 1 1 1

b _ a c

a^b '

draw the diagonally hatched rectangle (Fig. 6) with sides a b and
c d whose area is the number sought, (a b) (c d) , and which
is part of the rectangle with sides a and c . In order to obtain the former
rectangle from the latter, we take away first the horizontally hatched

rectangle a d, then the vertically
J L " hatched one b-c; in doing this we

have removed twice the double-hatched

rectangle b d , and we must put it back.

But these operations express precisely

the known formula

(2) (a b)(c d) = ac ad bc + bd.

As the most important psycholog-
ical moment to which the introduction
of negative numbers, upon this basis of

operations with letters, gave rise, that general peculiarity of human
nature shows itself, by virtue of which we are involuntarily inclined to
employ rules under circumstances more general than are warranted by the
special cases under which the rules were derived and have validity. This was
first claimed as a guiding principle in arithmetic by Hermann Hankel, in
his Theorie der komplexen Zahlsysteme* 1 , under the name "Prinzip von
der Permanenz der formalen Gesetze" **. I can recommend to your notice
this most interesting book. For the particular case before us, of transition
to negative numbers, the above principle would declare that one desired
to forget, in formulas like (1) and (2) the expressed assumptions as to
the relative magnitude of a and b and to employ them in other cases.
If one applies (2), for example, to a = c = 0, for which the formulas
were not proved at all, one obtains ( b) ( d) = + bd, i. e., the sign
rule for multiplication of negative numbers. In this manner we may
derive, in fact almost unconsciously, all the rules, which we must now

* Theory of Complex Number Systems.
1 Leipzig 1867.
** Principle of the permance of formal laws.

Negative Numbers. 27

designate, following the same line of thought, as almost necessary as-
sumptions, necessary insofar as one would have validity of the old rules
for the new concepts. To be sure, the old mathematicians were not happy
with this abstraction, and their uneasy consciences found expression in
names like invented numbers, false numbers, etc., which they gave to
the negative numbers on occasion. But , in spite of all scruples, the
negative numbers found more and more general recognition in the
sixteenth and seventeenth centuries, because they justified themselves
by their usefulness. To this end, the development of analytic geometry
without doubt contributed materially. Nevertheless the doubts per-
sisted, and were bound to persist, so long as one continued to seek for
a representation in the concept of a number of things, and had not
recognized the leading role of formal laws when new concepts are set
up. In connection with this stood the continually recurring attempts
to prove the rule of signs. The simple explanation, which was brought
out in the nineteenth century, is that it is idle to talk of the logical
necessity of the theorem, in other words, the rule of signs is not
susceptible of proof] one can only be concerned with recognizing the
logical permissibility of the rule, and, at the same time, that it is
arbitrary, and regulated by considerations of expedience, such as the
principle of permanence.

In this connection one cannot repress that oft recurring thought
that things sometimes seem to be more sensible than human beings.
Think of it: one of the greatest advances in mathematics, the intro-
duction of negative numbers and of operations with them, was not
created by the conscious logical reflection of an individual. On the
contrary, its slow organic growth developed as a result of intensive
occupation with things, so that it almost seems as though men had
learned from the letters. The rational reflection that one devised here
something correct, compatible with strict logic, came at a much later
time. And, after all, the function of pure logic, when it comes to setting
up new concepts, is only to regulate and never to act as the sole guiding
principle', for there will always be, of course, many other conceptual
systems which satisfy the single demand of logic, namely, freedom from
contradiction.

If you desire still other literature concerning questions about the
history of negative numbers, let me recommend Tropfkes Geschichte der
Elementarmathematik 1 * , as an excellent collection of material containing,
in lucid presentation, a great many details about the development of
elementary notions, views, and names.

1 Two volumes, Leipzig 1902/03- [Second edition revised and much enlarged, to
appear in seven volumes, of which six had appeared by 1924.] See also Cajori, F. (
History of Mathematics, Macmillan.

* History of Elementary Mathematics.

2g Arithmetic: The First Extension of the Notion of Number.

If we now look critically at the way in which negative numbers
are presented in the schools, we find frequently the error of trying to
prove the logical necessity of the rule of signs, corresponding to the
above noted efforts of the older mathematicians. One is to derive
(6) ( d) = +bd heuristically, from the formula (a b) (c d) and
to think that one has a proof, completely ignoring the fact that the
validity of this formula depends on the inequalities a > 6, c > d 1 . Thus
the proof is fraudulent, and the psychological consideration which would
lead us to the rule by way of the principle of permanence is lost in favor
of quasi-logical considerations. Of course the pupil, to whom it is thus
presented for the first time, cannot possibly comprehend it, but in the end
he must nevertheless believe it; and if, as it often happens, the repeti-
tion in a higher class does not supply the corrective, the conviction may
become lodged with some students that the whole thing is mysterious,
incomprehensible.

In opposition to this practice, I should like to urge you, in general,
never to attempt to make impossible proofs appear valid. One should
convince the pupil by simple examples, or, if possible, let him find out
for himself that, in view of the actual situation, precisely these con-
ventions, suggested by the principle of permanence, are appropriate in that
they yield a uniformly convenient algorithm, whereas every other convention
would always compel the consideration of numerous special cases. To be
sure, one must not be precipitate, but must allow the pupil time for
the revolution in his thinking which this knowledge will provoke. And
while it is easy to understand that other conventions are not advanta-
geous, one must emphasize to the pupil how really wonderful the fact
is that a general useful convention really exists ; it should become clear
to him that this is by no means self-evident.

With this I close my discussion of the theory of negative numbers
and invite you now to give similar consideration to the second extension
of the notion of number.

2. Fractions.

Let us begin with the treatment of fractions in the schools. There
the fraction a/b has a thoroughly concrete meaning from the start. In
contrast to the graphic picture of the integer, there has been only a
change of base: We have passed from the number of things to their
measure, from the consideration of countable things to measurable things.
The system of coins, or of weights, affords, with some restriction, and
the system of lengths affords completely, an example of measurable mani-
folds. These are the examples with which the idea of the fraction is

1 See, for example, . Heis, Sammlung von Beispielen und Aufgaben aus der
Arithmetik und Algebra. Edition 1904, p. 46, 106108.

Fractions. 29

given to every pupil. No one has great difficulty in grasping the meaning
of x / 3 meter oder x / 2 pound. The relations =,>,<, between fractions
can be immediately developed by means of the same concrete intuition,
and likewise the operations of addition and subtraction, as well as the
multiplication of a fraction by an integer. After this, general multiplication
can easily be made ofeprehensible : To multiply a number by a/b means
to multiply it by a ana then to divide by 6; in other words: the product is
derived from the multiplicand just as a/b is derived from \ . Division by
a fraction is then presented as the operation inverse to multiplication:
a divided by 2/3 is the number which multiplied by 2/3 gives a. These
notions of operations with fractions combine with that of negative
numbers so that one finally has the totality of all rational numbers.
I cannot enter into the details of this building-up process, which, in the
school, takes, of course, a long time. Let us rather compare it at once
with the perfected presentation of modern mathematics, using for
this purpose the above mentioned books of Weber- Wellstein and
Burkhardt 1 .

Weber-Wellstein emphasizes primarily the formal point of view which,
from the multiplicity of possible interpretations, selects what is of
necessity common to all. According to this view, the fraction a/b is
a symbol, a "number-pair" with which one can operate according to
certain rules. These rules, which in our discussion above arose naturally
from the meaning of fraction, have here the character of arbitrary con-
ventions. For example, that which, to the pupil, is an obvious theorem
concerning the multiplication or division of both terms of a fraction
by the same number, appears here as a definition of equality: two
fractions a/b, c/d are called equal when ad = be. Similarly, greater than

and smaller than are defined, and one agrees that the fraction ( j--j )

shall be called the sum of the two fractions a/b, c/d, etc. It is thus proved
that the operations, so defined in the new domain of numbers, possess
formally exactly the properties of addition and multiplication for in-
tegers, i. e., they satisfy the eleven fundamental laws which have been
repeatedly enumerated.

Burkhardt does not proceed quite so formally as does Weber-Well-
stein, whose presentation we have sketched in its essentials. He looks
upon the fraction a/b as a sequence of two operations in the domain of
integers: a multiplication by a and a division by b, in which the object
upon which these operations are performed is an arbitrarily chosen
integer. If one undertakes two such "pairs bf operations" a/b, c/d, this
is said to correspond to multiplication of the fractions, and one sees easily
that the operation so resulting is none other than multiplication by a c
and division by b d, so that the rule for the multiplication of fractions,

1 In what follows, the first editions of these books have been used.

Arithmetic: The First Extension of the Notion of Number.

y) " \~d) ~ \b^~d)' ^toincd ou ^ f ^ e c ^ ear meaning of the fractions,
but not determined merely as an arbitrary convention. One can, of
course, treat division in the same way. Addition and subtraction, on
the other hand, do not admit of such a simple explanation with this

representation ; thus the formula -j- + -T- = , ,^ remains, with Burk-

hardt also, only a convention for which he adtfuces only reasons of
plausibility.

Let us now compare the older presentation in the schools, with the
modern conception just sketched. According to the latter, in the one
book as well as in the other, we are left really completely in the field of
integers, in spite of the extension of the notion of number. It is merely
assumed that the totality of whole numbers is intuitively grasped, or
that the rules of operation with them are known; the things newly
defined as number-pairs, or as operations with whole numbers, fit
completely into this frame. The school treatment, on the other hand,
is based entirely on the newly acquired conception of measurable quan-
tities, which supplies an immediate intuitive picture of fractions. We
can best grasp this difference if we imagine a being who has the notion
of whole numbers, but no conception of measurable quantities. For him
the school presentation would be wholly unintelligible, whereas he could
well comprehend the discussions of either Weber- Wellstein or Burkhardt.

Which of the two methods is the better? What does each accomplish?
The answer to this will be like the one we gave recently when we put
the analogous question concerning the different conceptions of integers.
The modern presentation is surely purer, but it is also less rich. For,
of that which the traditional curriculum supplies as a unit, it gives
really only one part : the abstract and logically complete introduction
of certain arithmetic concepts, called "fractions" , and of operations with
them. But it leaves unexplained an entirely independent and no less
important question: Can one really apply the theoretical doctrine so
derived, to the concrete measurable quantities about us? Again one
could call this a problem of "applied mathematics", which admits an
entirely independent treatment. To be sure, it is questionable whether
such a separation would be desirable pedagogically. In Weber- Wellstein,
moreover, this splitting of the problem into two parts finds characteristic
expression. After the abstract introduction of operations with fractions,
of which alone we have thus far taken account, they devote a special
(the fifth) division called "ratios" to the question of applying rational
numbers to the external world. The presentation is, to be sure, rather
abstract than intuitive.

I shall now close this discussion of fractions with a general remark
concerning the totality of rational numbers, where, for the sake of
clearness, I shall make use of the representation upon a straight line.

Irrational Numbers. if

Think of all points with rational abscissas marked upon this line; we
designate them briefly as rational points. We say, then, that the totality
of these rational points on the axis of abscissas is "dense", meaning
that in every interval, however small, there are still infinitely many

. . . I . . . . . I I , . , , , I , ,

rational points. If we wish to avoid putting anything new into the
notion of rational numbers, we might say, more abstractly, that between
any two rational points there is always another rational point. It follows
that one can separate from the totality of rational points, finite parts
which contain neither a smallest nor a largest element. The totality
of all rational points between and 1 , these points excluded, is an
example. For, given any number between and 1 , there would still
be a number between it and 0, i. e., a smaller, and a number between
it and 1 , i. e., a larger. In their systematic development, these concepts
belong to the theory of point sets of Cantor. In fact, we shall make use
later of the totality of rational numbers, together with the property
just mentioned, as an important example of a point set.

I shall pass now to the third extension of the number system: the
irrational numbers.

3. Irrational Numbers.

Let us not spend any time in discussing how this field is usually
treated in the schools, for there one does not get much beyond a few
examples. Let us rather proceed at once to the historical development.
Historically, the origin of the concept of irrational
numbers lies certainly in geometric intuition and in
the requirements of geometry. If we consider, as
we did just now, that the set of rational points is
dense on the axis of abscissas, then there are still

other points on it. Pythagoras is said to have shown p .

this in a manner somewhat as follows. Given a right

triangle with each leg of length 1 , then the hypotenuse is of length

}/2, and this is certainly not a rational number; for if one puts y 2 = -r

where a and b are integers, prime to each other, one is led easily by the
laws of divisibility of integers to a contradiction. // we now lay off
geometrically on the axis of abscissas, beginning at zero, the segment thus
constructed, we obtain a non-rational point which is not one of the original
set that is dense on the axis. Furthermore, the Pythagoreans certainly
were aware that, in most cases, the hypotenuse, 1/m 2 + n 2 , of a right
triangle with legs m and n t is irrational. The discovery of this extra-
ordinarily essential fact was indeed worth the sacrifice of one hundred

12 Arithmetic: The First Extension of the Notion of Number.

oxen with which Pythagoras is said to have celebrated it. We know, also
that the Pythagorean School was fond of searching out those special pairs
of values for m and n for which the right triangle has three commensurable
sides, whose lengths, in an appropriately selected unit of measure, can
be expressed in integers (so called Pythagorean numbers). The simplest
example of one of these number-triples is '3, .4&5

Later Greek mathematicians studied, in addition to these simplest
irrationalities, others that were more complicated; thus one finds in

Euclid types such as }//0 + /&, and the like. We may say, however,
in general, that they confined themselves essentially to such irrationali-
ties as one obtains by repeated extraction of square root, and which
can therefore be 1 constructed geometrically with ruler and compasses.
The general idea of irrational number was not yet known to them.

I must, modify this remark somewhat, however, in order to avoid
misunderstanding. The more precise statement is that the Greeks
possessed no method for producing or defining, arithmetically, the
general irrational number in terms of rational numbers. This is a result
of modern development and will soon engage our attention. Nevertheless,
from another point of view they were familiar with the notion of the
general real number which was not necessarily rational; but the concept
had an entirely different appearance to them because they did not use
letters for general numbers. In fact they studied, and Euclid developed
very systematically, ratios of two arbitrary segments. They operated with
such ratios precisely as we do today with arbitrary real numbers. In-
deed we find in Euclid definitions which suggest strongly the modern
theory of irrational numbers. Moreover the name used is different from
that of the natural number; the latter is called ciQiftfios , whereas the
line ratio, the arbitrary real number, is called Myos.

I should like to add a remark concerning the word "irrational". It
is without doubt the translation into Latin of the Greek "fiyloj'og".
The Greek word, however, meant presumably "inexpressible" and im-
plied that the new numbers, or line ratios, could not, like the rational
numbers, be expressed by the ratio of two whole numbers 1 . The
misunderstanding put upon the Latin "ratio", that it could convey
only the meaning "reason", gave to "irrational" the meaning "unreaso-
nable", which seems still to cling to the term irrational number.

The general idea of the irrational number appeared first at the end
of the sixteenth qentury as a consequence of the introduction of decimal
fractions, the use of which became established at that time in connection
with the appearance of logarithmic tables. If we transform a rational
number into a decimal, we may obtain infinite decimals*, as well as finite

1 See Tropfke, second edition, Vol. 2, p. 71.

2 For complete treatment of this subject see, p. 40 et seq.

Irrational Numbers.

decimals, but they will always be periodic. The simplest example is
J = 0.333 . . . , i.e. , a decimal whose period of one digit begins imme-
diately after the decimal point. Now there is nothing to prevent our
thinking of an aperiodic decimal whose digits proceed according to any
definite law whatever, and anyone would instinctively consider it as
a definite, and hence|& non-rational, number. By this means the general
notion of irrational number is established. It arose to a certain extent
automatically, by the consideration of decimal fractions. Thus, histori-
cally, the same thing happened with irrational numbers that, as we have
seen, happened with negative numbers. Calculation forced the intro-
duction of the new concepts, and without being concerned much as to
their nature or their motivation, one operated with them, the more
particularly since they often proved to be extremely useful.

It was not until the sixth decade of the nineteenth century that the
need was felt for a more precise arithmetic formulation of the foun-
dations of irrational numbers. This occurred in the lectures which
Weierstrass delivered at about that date. In 1872, a general foundation
was laid simultaneously by G. Cantor of Halle, the founder of the theory
of point sets, and independently by R. Dedekind of Braunschweig. I
will explain Dedekind' s point of view in a few words. Let us assume
a knowledge of. the totality of rational numbers, but let us exclude
all space perception, which would force upon us forthwith the notion of
the continuity of the number series. With this understanding, in order
to attain to a purely arithmetic definition of the irrational number,
Dedekind sets up the notion of a "cut" in the domain of rational numbers.
If r is any rational number, it separates the totality of rational numbers
into two parts A and B such thai ev$ry number in A is smaller than any
number in B and every rational number belongs to one of these two classes.
A is the totality of all rational numbers which are smaller than r , B those
that are larger, whereby r itself may be thought of indifferently as be-
longing to the one or to the other. Besides these "proper cuts" there are
also "improper cuts", these being separations of all rational numbers
into two classes having the same properties except that they are not
brought about by a rational number, i. e., separations such that there
is neither a smallest rational number in B nor a largest in A . An example

of such an improper cut is supplied by, say, "^2 = 1.414 ... In fact,
every infinite decimal fraction defines a cut, provided one assigns to B
every rational number which is larger than every approximation to the
infinite decimal, and to A every other rational number ; each number
in A would thus be equalled or exceeded by at least one approximation
(and hence by infinitely many). One can easily show that this cut is
proper if the decimal is periodic, improper if it is not periodic.

With these considerations as his basis, Dedekind sets up his definition,
which, from a purely logical standpoint, must be looked upon as an

Klein, Elementary Mathematics. 3

^4 Arithmetic: The First Extension of the Notion of Number.

arbitrary convention : A cut in the domain of rational numbers is called
a rational number or an irrational number according as the cut is proper
or improper. A definition of equality follows from this at once: Two
numbers are said to be equal if they yield the same cut in the domain of
rational numbers. From this definition we can immediately prove for
example, that, J / 3 is equal to the infinite decimal 0.^333 .... If we accept
this standpoint, we must demand a proof, i. e., a process of reasoning
depending upon the definition given, although this would appear quite
unnecessary to one approaching the subject naively. Moreover, such
a proof is immediate, if one reflects that every rational number smaller
than x / 3 will be exceeded ultimately by the decimal approximations,
whereas these are smaller than every rational number which exceeds J.
The corresponding definition in the lectures of Weierstrass appears in
the following form: Two numbers are called equal if they differ* by less
than any preassigned constant, however small. The connection with the
preceding explanation is clear. The last definition becomes striking if
one reflects why 0.999 is equal to 1 ; the difference is certainly
smaller than 0.1, smaller than 0.01, etc., that is, it is exactly zero,
according to the definition.

If we enquire how it happens that we can admit the irrational
numbers into the system of ordinary numbers and operate with them
in just the same way, the answer is to be found in the validity of the
monotonic law for the four fundamental operations. The principle is as
follows: // we wish to perform upon irrational numbers the operation of
addition , multiplication, etc., we can enclose them between ever narrowing
rational limits and perform upon these limits the desired operations ; then,
because of the validity of the monotonic law, the result will also be enclosed
between ever narrowing limits.

It is hardly necessary for me to explain these things in greater
detail, since very readable presentations of them are easily available in
many books, especially in Weber-Wellstein and in Burkhardt. I hope
that you will read more fully than I could tell you here in these books,
about the definition of irrational numbers.

I should prefer, rather, to talk about something which you will
hardly find in the books, namely, how, after establishing this arithmetic
theory, we can pass to the applications in other fields. This applies in
particular, to analytic geometry, which to the naive perception appears
to be (and psychologically really is) the source of irrational numbers.
If we think of the axis of abscissa, with the origin and also the rational
points marked on it, as above, then these applications depend upon
the following fundamental principle: Corresponding to every rational or
irrational number there is a point which has this number as abscissa and,
conversely, corresponding to every point on the line there is a rational or
an irrational number, viz., its abscissa. Such a fundamental principle,

Irrational Numbers.

which stands at the head of a branch of knowledge, and from. which
all that follows is logically deduced, while it itself cannot be logically
proved, may properly be called an axiom. Such an axiom will appear
intuitively obvious or will be accepted as a more or less arbitrary con-
vention, by each person according to his gifts. This axiom concerning
the one-to-one correspondence between real numbers on one hand, and
the points of a straight line on the other, is usually called the Cantor
axiom because G. Cantor was the first to formulate it specifically (in
the Mathematische Annalen, vol. 5, 1872).

This is the proper place to say a word about the nature of space
perception. It is variously ascribed to two different sources of knowledge.
One the sensibly immediate, the empirical intuition of space, which we
can control by means of measurement. The other is quite different,
and consists in a subjective idealizing intuition, one might say, perhaps,
our inherent idea of space, which goes beyond the inexactness of sense
observation. I pointed out to you an analogoiis difference when we were
discussing the notion of number. We may characterize it best as follows:
It is immediately clear to us what a small number means, like 2 or 5,
or even 7, whereas we do not have such immediate intuition of a larger
number, say 2503- Immediate intuition is replaced here by the sub-
jective intuition of an ordered number series, which we derive from
the first numbers by mathematical induction. There is a similar situation
regarding space perception. Thus, if we think of the distance between
two points, we can estimate or measure it only to a limited degree of
exactness, because our eyes cannot recognize as different two line-segments
whose difference in length lies below a certain limit. This is the concept
of the threshold of perception which plays such an important role in
psychology. This phenomenon still persists, in its essentials, when we
aid the eye with instruments of the highest precision; for there are
physical properties which prohibit our exceeding a certain degree of
exactness. For instance, optics teaches that the wave-length of light,
which varies with the color, is of the order of smallness of 1 / 10 oo mm -
(= 1 micron); it shows also that objects whose dimensions are of this
order of smallness cannot be seen distinctly with the best microscopes
because diffraction enters then and hence no optical image can give
exact reproductions of the details. The result of this is the impossibility, by
direct optical means, of getting measures of length that are finer than to
within one micron, so that, when measured lengths are given in millimeters,
only the first three decimals can have an assured meaning. In the same
way, in all physical observations and measurements, one meets such
threshold values which cannot be passed, which determine the extreme
limits of possible exactness of lengths which have been measured and
expressed in millimeters. Statements beyond this limit have no meaning,
and are an evidence of ignorance or of attempted deception. One often

3 () Arithmetic: The First Extension of the Notion of Number.

finds such excessively exact numbers in the advertisements of medicinal
springs, where the percentage of salt, which really varies with the
time, is given to a number of decimal places which could not possibly
be determind by weighing.

In contrast with this property of empirical space perception which
is restricted by limitations on exactness, abstract, or ideal space perception
demands unlimited exactness, by virtue of which, in view of Cantor's axiom,
it corresponds exactly to the arithmetic definition of the number concept.

In harmony with this division of our perception, it is natural to
divide mathematics also into two parts, which have been called mathe-
matics of approximation and the mathematics of precision. If we desire
to explain this difference by an interpretation of the equation / (x) = 0,
we may note that, in the mathematics of approximation, just as in our
empirical space perception, one is not concerned that / (x) should be
exactly zero, but .merely that its absolute value |/ (x)\ should remain
below the attainable threshold of exactness . The symbol / (x) = is
merely an abbreviation for the inequality | / (x) \ <C e , with which one
is really concerned. It is only in the mathematics of precision that one
insists that the equation / (x) = be exactly satisfied. Since mathe-
matics of approximation alone plays a r61e in applications, one might
say, somewhat crassly, that one needs only this branch of mathematics,
whereas the mathematics of precision exists only for the intellectual
pleasure of those who busy themselves with it, and to give valuable
and indeed indispensable support for the development of mathematics
of approximation.

In order to return to our real subject, I add here the remark that
the concept of irrational number belongs certainly only to mathematics of
precision. For, the assertion that two points are separated by an ir-
rational number of millimeters cannot possibly have a meaning, since,
as we saw, when our rigid scales are measured in meters, all decimal
places beyond the sixth are devoid of meaning. Thus in practice we can,
without concern, replace irrational numbers by rational ones. This may
seem, to be sure, to be contradicted by the fact that, in crystallography,
one talks of the law of rational indices, or by the fact that in astronomy,
one distinguishes different cases according as the periods of revolution
of two planets have a rational or an irrational ratio. In reality, however,
this form of expression only exhibits the many-sidedness of language ;
for one is using here rational and irrational in a sense entirely different
from that hitherto used, namely, in the sense of mathematics of approxi-
mation. In this sense, one says that two magnitudes have a rational
ratio when they are to each other as two small integers, say 3/7; whereas
one would call the ratio 2021/7053 irrational. We cannot say how large
numerator and denominator in this second case must be, in general,
since that depends upon the problem in hand. I discussed all these

Number Theory in the Schools. 17

interesting relations in a course of lectures in the Summer Semester
of i90i, wmcn was lithographed in 1902 and which will constitute the
third volume of the present work (see the preface to the third edition,
p. V) : Applications of Differential and Integral Calculus to Geometry,
a Revision of Principles [Elaborated by C. H. Muller].

In conclusion let me say, in a few words, how I would have these
matters handled in the schools. An exact theory of irrational numbers
would hardly be adapted either to the interest or to the power of com-
prehension of most of the pupils. The pupil will usually be content
with results of limited exactness. He will look with astonished approval
upon correctness to within 1 / 1000 mm and will not demand unlimited
exactness. For the average pupil it will be sufficient if one makes the
irrational number intelligible in general by means of examples, and
this is what is usually done. To be sure, especially gifted individual
pupils will demand a more complete explanation than this, and it will
be a laudable exercise of pedagogical skill on the part of the teacher
to give such students the desired supplementary explanation without
sacrificing the interests of the majority.

III. Concerning Special Properties of Integers

We shall now begin a new chapter which will be devoted to the
actual theory of integers, to the theory of numbers, or arithmetic in its
narrower sense. I shall first recall in tabular form the individual ques-
tions from this science which appear in the school curriculum.

1 . The first problem of the theory of numbers is that of divisibility :
Is one number divisible by another or not?

2. Simple rules can be given which enable us easily to decide as to
the divisibility of any given number by smaller numbers, such as 2, 3, 4,
5, 9, 11, etc.

3. There are infinitely many prime numbers, that is, numbers which
have no integral divisors except one and themselves) : 2, 3, 4, 5, 9, 11, etc.

4. We are in control of all of the properties of given integers if we
know their decomposition into prime factors.

5. In the transformation of rational fractions into decimal fractions
the theory of numbers plays an important role ; it shows why the decimal
fraction must be periodic and how large the period is.

Although such questions may be considered in secondary schools,
when the pupils are' between the ages of eleven and thirteen, the theory
of numbers comes up only in isolated places during the later years,
and, at most, the following points are considered.

6. Continued fractions are taught occasionally, although not in all
schools.

7. Sometimes instruction is given also in Diophantine equations, that
is, equations with several unknowns which can take only integral values.

38 Arithmetic: Concerning Special Properties of Integers.

The Pythagorean numbers of which we spoke (see p. 32), furnish an
example; here one has to do with triplets of integers which satisfy the
equation

a * + b* = c*.

8. The problem of dividing the circle into equal parts is closely related
to the theory of numbers, although the connection is hardly ever worked
out in the schools. If we wish to divide the circle into n equal parts,
using, of course, only ruler and compasses, it is easy to do it for n = 2, 3>
4, 5,6. It cannot be done, however, if n = 7, hence we stop respect-
fully when we come to this problem in the school. To be sure, it is not
always stated definitely that -this construction is really impossible when
n = 7, a fact whose explanation lies somewhat deep in number-theo-
retic considerations. In order to forestall misunderstandings, which un-
fortunately often arise, let me say, with emphasis, that one is concerned
here again with a problem of mathematics of precision, which is devoid
of meaning for the applications. In practice, even in cases where an
"exact" construction is possible, it would not be used ordinarily; for,
in the field of mathematics of approximation, the circle can be divided
into any desired number of equal parts more suitably by simple skillful
experiment ; and any prescribed, practically possible, degree of exactness
can be attained. Every mechanician who makes instruments that carry
divided circles proceeds in this way.

9. The higher theory of numbers is touched by the school curriculum
in one other place, namely, when n is calculated, during the study of the
quadrature of the circle. We usually determine the first decimal places
for &, by some method or other, and we mention incidentally, perhaps,
the modern proof of the transcendence of n which sets at rest the old problem
of the quadrature of the circle with ruler and compasses. At the end of
this course I shall consider this proof in detail. For the present I shall
give merely a prescise formulation of the fact, namely, that the number
n does not satisfy any algebraic equation with integral coefficients:

l + ... +kji+ \ -0.

It is especially important that the coefficients be integers, and it is for
this reason that the problem belongs to the theory of numbers. Of
course here, again, one is concerned solely with a problem of the mathe-
matics of precision, because it is only in this sense that the number-
theoretic character of n has any significance. The mathematics of
approximation is satisfied with the determination of the first few
decimals, which permit us to effect the quadrature of the circle with
any desired degree of exactness.

I have sketched for you the place of the theory of numbers in
the schools. Let us consider now its proper place in university instruction
and in scientific investigation* In this connection I should like to divide

Number Theory in the University. ^9

research mathematicians, according to their attitude toward theory of
numbers, into two classes, which I might call the enthusiastic class and
the indifferent class. For the former there is no other science so beautiful
and so important, none which contains such clear and precise proofs,
theorems of such impeccable rigor, as the theory of numbers. Gauss
said "If mathematics is the queen of sciences, then the theory of numbers
is the queen of mathematics" . On the other hand, theory of numbers
lies remote from those who are indifferent; they show little interest in
its development, indeed they positively avoid it. The majority of
students might, as regards their attitude, be put into the second class.

I think that the reason for this remarkable division can be summarized
as follows: On the one hand the theory of numbers is fundamental for
all more thoroughgoing mathematical research] proceeding from entirely
different fields, one comes at last, with extraordinary frequency, upon
relatively simple arithmetic facts. On the other hand, however, the
pure theory of numbers is an extremely abstract thing, and one does not
often find the gift of ability to understand with pleasure anything so
abstract. The fact that most textbooks are at pains to present the sub-
ject in the most abstract way tends to accentuate this unattractiveness
of the subject. I believe that the theory of numbers would be made more
accessible, and would awaken more general interest, if it were presented
in connection with graphical elements and appropriate figures. Although
its theorems are logically independent of such aids, still one's compre-
hension would be helped by them. I attempted to do this in my lectures
in 1895/96 1 and a similar plan is followed by H. Minkowski in his book
on Diophantische Approximationen 2 . My lectures were of a more ele-
mentary introductory character, whereas Minkowski considers at an
early point special problems in a detailed manner.

As to textbooks in the theory of numbers, you will often find all you
need in the textbooks in algebra. Among the large number of books
on the theory of real numbers, I would mention especially Bachman's
Grundlagen der neueren Zahlentheorie*.

In the more special number -theoretic discussions which I shall give
here, I shall keep touch with the points mentioned above and I shall
endeavor especially to present the matter as graphically as possible,
While I shall restrict myself to material that is valuable for the teacher,
I shall by no means put it into a form suitable for immediate presentation
to the pupils. The necessity for this arises from my experiences in

1 Ausgew&hltes Kapitel der Zahlentheorie (mimeographed lectures written up
by A. Sommerfeld and Ph. Furtwangler) . Second printing (already exhausted).
Leipzig 1907.

2 With an appendix: Fine Einfuhrung in die Zahlentheorie. Leipzig 1907-
8 Sammlung Schubert No. 53. Leipzig 1907. [Second edition published by

R. Hauszner 1921.] See also Carmichael, R. D., Theory of Numbers. Wiley.

40 Arithmetic: Concerning Special Properties of Integers.

examinations, which show me that the number-theoretic information of
candidates is often confined to catchwords which have no thorough
knowledge back of them. Every candidate can tell me that n is "trans-
cendental" ; but many of them do not know what that means; I was
told, once, that a transcendental number was neither rational nor ir-
rational. Likewise I often find candidates who tell me that the number
of primes is infinite, but who have no notion as to the proof, although
it is so simple.

I shall start my number-theoretic discussion with this proof, assuming
that you are acquainted with the first two points metioned in our list.
As a matter of history I remind you that this proof was handed on to
us by Euclid, whose "elements" (Greek oroi%ia) contained not only
his system of geometry, but also algebraic and arithmetic information in
geometric language. Euclid's transmitted proof of the existence of in-
finitely 'many prime numbers is as follows : Assuming that the sequence
of prime numbers is finite, let it be 1 , 2, 3 , 5 , . . . , p\ then the number
N = (1 2- 3 5 - p) 1 is not divisible by any of the numbers
2 , 3 > 5 , . . p since there is always the remainder 1 ; hence N must
either itself be a prime number or there are prime numbers larger than p .
Either of these alternatives contradicts the hypothesis, and the proof
is complete.

In connection with the fourth point, the separation into prime factors,
I should like to call to your attention one of the older factor tables:
Chernac, Cribum Arithmeticum 1 , a large, meritorious work which de-
serves, historically, all the more attention because it is so reliable. The
name of the table suggests the sieve of Eratosthenes. The idea on which
it was based is that we should discard gradually from the series of all
integers those which are divisible by 2,3,5,-.., so that only the
prime numbers would remain. Chernac gives the decomposition into
prime factors of all integers up to 1020000 which are not divisible
by 2, 3, or 5; all the prime numbers are marked with a bar. It
was in the Chernac work that all the prime numbers lying within
the limits stated above were first given. During the nineteenth century
the determination was extended to all prime numbers as far as nine
million.

I turn now to the fifth point, the transformation of ordinary fractions
into decimal fractions. For the complete theory I shall refer you to Weber-
Wellstein, and I shall explain here only the principle of the method by
means of a typical example. Let us consider the fraction \jp t where p
is a prime number different from 2 and 5- We shall show that \jp is equal
to an infinite periodic decimal, and that the number d of places in the
period is the smallest exponent for which 10* 5 , when divided byp, leaves \

1 Deyenter 1811.

Prime Numbers.

as a remainder, or that, in the language of number theory, 6 is the
smallest exponent which satisfies the "congruence" :

The proof requires, in the first place, the knowledge that this congruence
always has a solution. This is supplied by the theorem of Fermat, which
states that for every prime number p except 2 and 5:

We shall omit here the proof of this fundamental theorem, which is
one of the permanent tools of every mathematician. Secondly, we must
borrow from the theory of numbers the theorem that the smallest
exponent in question, (3, is either p 1 itself or a divisor of p \ . We

can apply this to the given value p and find that - is an integer N
so that one has:

-!5? = ^ + N

P P^

If we now think of \tfjp, as well as l//>, converted into a decimal,
the digits in the two decimals must be identical, since the difference
is an integer. But since \tf/p is got from \\p by moving the decimal
point 6 places to the right, it follows that the digits in the decimal
expression of \/p are unaltered by this operation, in other words that
the decimal fraction \lp consists of continued repetition of the same "period"
of d digits.

In order now to see that there cannot be a smaller period of 8' < <5
digits one needs only to prove that the digit number 6' of every period
must satisfy the congruence 10* 5 ' = 1 ; for we know that 6 was the
smallest solution of this congruence. This proof will result if we pursue
the preceding argument in the reverse direction. It follows from our
assumption that 1/p and \tf'jp coincide in their decimal places, hence

10^' 1 A'

that --- - is an integer N', and therefore that 10 1 is divisible by p ,

or, in other words, that 10* 5 ' = 1 (mod^). This completes the proof.
I will give you a few of the simplest instructive examples, which will
show that d can take widely different values, both smaller than and
equal to p 1 . Notice first that for:

4 = 0.333...

the number of digits in the period is 1 , and that in fact, 10 1 = 1 (mod 3)'
Similarly we find

^ = 0.0909...,

whence d = 2, and correspondingly 10 1 slo,10 2 sl (mod 11). The
maximum value = p 1 appears in the example:

| -0,142857142857... .

42 Arithmetic: Concerning Special Properties of Integers.

Here <J = 6 and we have, in fact, 10 1 = 3 , 10 2 = 2, 10 3 = 6, 10 4 = 4,
10 5 = 5 , and 10 6 = 1 (mod 7).

Now let us take up, in a similar way, the sixth point of my list,
continued fractions. I shall not present this, however, in the usual
abstract arithmetic manner, since you will find it given elsewhere, e. g.,
in Weber-Wellstein. I shall take this opportunity to show you how
number-theoretic things take on a clear and easily intelligible form
through geometric and graphical presentation. In this use of geometric
aids in number theory we are really only retracing the steps followed
by Gauss and Dirichlet. It was the later mathematicians, say from i860
on, who banished geometric methods from the theory of numbers. Of
course, I can give here only the most important trains of thought and
theorems, without proof, and I shall assume that you are not entire
strangers to the elementary theory of continued fractions. My litho-
graphed lectures on number theory 1 contain a thoroughgoing account.

You know how the development of a given positive number co into a
continued fraction arises. We separate out the largest positive integer n Q
contained in co and write:

a) = n + r , where ^ r Q < 1 ,

then, if r Q =f= 0, we treat l/r as we did co :

l/^o = n \ + r i> where < ^ < 1 ,

and continue in the same way:

\\r^ = n 2 + r 2 , where g r 2 < \,
\jr z = n s + r 3 , where < r 3 < 1 ,

The process terminates after a finite number of steps if co is rational,
because a vanishing remainder r v must appear in that case ; otherwise
the process goes on indefinitely. In any case, we write, as the development
of (o into a continued fraction :

As an example, the continued fraction for n is
^ = 3-14159265 =3 + -

l + l

292 +,

1 See also Klein, F., Gesammelte Mathematische Abhandlungen, Vol. II, pp.209
to 211.

Continued Fractions.

If we stop the development after the first, second, third, . . . partial
denominator, we obtain rational fractions, called convergents:

these give remarkably good approximations to the number co , or, to
speak more exactly, each one of them gives an approximation which is
closer than that given by any other rational fraction which does not have
a larger denominator. Because of this property, continued fractions are
of practical importance where one seeks the best possible approximation
to an irrational number, or to a fraction with a large denominator (e. g.
a many-place decimal) by means of a fraction having the smallest
possible denominator. The following convergents of the continued frac-
tion for a, converted into decimals, enable one to see how close the
approximations are to the value n = 3,14159265 . . .'

You will observe, moreover, in this example, that the convergents are
alternately less than and greater than n. This is true in general, as is
well known, that is the successive convergents of the continued fraction
for oj are alternately less than and greater than co , and enclose it between
ever narrowing limits.

Let us now enliven these considerations with geometric pictures.
Confining our attention to positive numbers, let us mark all those points
in the positive quadrant of the %y plane (see Fig. 8) which have integral
coordinates, forming thus a so called point lattice. Let us examine this
lattice, I am tempted to say this "firmament" of points, with our point
of view at the origin. The radius vector from to the point (x = a,
y = b) has for its equation

x _ a
7 = T '

and conversely, there are upon every such ray, x/y = &, where A = a/b is
rational, infinitely many integral points (ma, mb), where m is an arbi-
trary whole number. Looking from 0, then, one sees points of the
lattice in all rational directions and only in such directions. The field of
view is everywhere "densely" but not completely and continuously filled
with "stars". One might be inclined to compare this view with that
of the milky way. With the exception of itself there is not a single
integral point lying upon an irrational ray x/y = ft) , where ft) is irrational,
which is very remarkable. If we recall Dedekind's definition of irrational
number, it becomes obvious that such a ray makes a cut in the field

Arithmetic: Concerning Special Properties of Integers.

of integral points by separating the points into two point sets, one lying
to the right of the ray and one to the left. If we inquire how these
point sets converge toward our ray x/y = co , we shall find a very simple
relation to the continued fraction for co . By marking each point (x = p v ,
y = q^ t corresponding to the convergent p v jq v , we see that the rays
to these points approximate to the ray x]y = co better and better, alter-
nately from the left and from the right,
just as the numbers p v /q v approxi-
mate to the number co. Moreover, if
one makes use of the known number-
theoretic properties of p v , q v , one
finds the following theorem : Imagine
pegs or needles affixed at all the integral
points, and wrap a tightly drawn string
about the sets of pegs to the right and to
the left of the co-ray , then the vertices
of the two convex string-polygons which
bound our two point sets will be precisely
the points (p v , q^ whose coordinates are
the numerators and denominators of the
successive conver gents to co , the left poly-
gon having the even convergents, the
Fig. 8. right one the odd. This gives a new,

and, one may well say, an extremely

graphic definition of a continued fraction. The representation in Fig. 8
corresponds to the example

CO =

1+1

1 +1

1 +

which is the irrationality associated with the regular decagon. In this
example, the first few vertices of the two polygons are

left: ft = o, ? = 1 ; ft = 1 , 9 2 = 2; ft = 3, fc = 5; . . .
right: p l = 1 , q l = 1 ; ft = 2, ? 3 = 3; ft = 5, ? 5 = 8; . . .

The values p v , q v for n grow much more rapidly, so that one could
hardly draw the corresponding representation. The proof of our theorem,
which I cannot give here, can be found in detail on page 43 of in my
lithographed lectures.

I shall now pass on to the treatment of the seventh point, the Pytha-
gorean numbers, where we shall use space perception in a somewhat
different form. Instead of the equation:

m 2 + 6 2 = c 2 .

Pythagorean Numbers. 45

whose integral solutions are sought, let us set:

(2) */c = f, 6/c = ij
and consider the equation:

(3) P + i = 1 ,

with the problem of finding all the rational number-pairs , rj which
satisfy it. Accordingly, we start from the representation of all rational
points , r) (i.e. all points with rational coordinates f, rj), which will
fill the ^-plane "densely". I 2 + *? 2 = 1 is the equation of the
circle about the origin in this plane. It is our
task to see how this circle threads its way through
the dense set of rational points, in particular, to
see which of these points it contains. We know a
few such points of old, such as the intercepts
with the axes, one of which, S ( = 1, q = 0) ,
we shall consider (see Fig. 9). All rays through
5 are given by the equation

(4) ? = *(* + !); Fig ' 9 -

we call such a ray rational or irrational according as the parameter A is
rational or not. We have now the double theorem that every rational
point of the circle is projected from S by a rational ray and that every rational
ray (4) meets the circle in a rational point. The first half of the theorem
is obvious. We prove the second half by substituting from (4) in (3).
This gives for the abscissas of the points of intersection the equation

We know one solution of this equation, = 1 , which corresponds to
the intersection S; for the other, one gets by easy calculation

(5aj * = f^

and from (4) the corresponding ordinate

(5b) ^ = TTF-

From (5 a) and (5b) it follows that the second intersection is a rational
point if A is rational.

Our double theorem, now fully proved, can be stated also as follows.
All the rational points of the circle are represented by formulas (5) -if A is
an arbitrary rational number. This solves our problem and we need only
to transform to whole numbers. For this purpose we put

;. = n/m ,

46 Arithmetic: Concerning Special Properties of Integers.

where n,m are integers and obtain from (5):

_ m 2 n 2 _ 2m n

f - m 2 + w a ' ^ - m 2 ~+ V 2 '

as the totality of rational solutions of (3). All integral solutions of the
original equation (1), i.e., all Pythagorean numbers are therefore given by
the equations

a m 2 w 2 , b = 2 wn, c = m 2 + w 2 ;

obtains the totality of solutions which have no common divisor if m
and n take all pairs of relatively prime integral values. We have thus a
graphic deduction of a result which usually appears very abstract.
In this connection I should like to discuss the great Fermat. theorem.
It is quite after the manner of the geometers of, antiquity that one
should generalize the question regarding Pythagorean numbers, from
the plane to space of three and more dimensions in the following manner.
Is it possible that the sum of the cubes of two integers should be a cube?
Or that the sum of two fourth powers should be a fourth power, etc.?
In general, has the equation

x n + y n = z n ,

where n is an arbitrary integer, solutions which are whole numbers'? To
this question Fermat gave the answer no, in the theorem named after
him : The equation % n + y n = z n has no integral solutions for integral values
of n except when n = \ and n = 2 . Let me begin with a few historical
notes. Fermat lived from 1601 to 1665 and was a parliamentary coun-
cillor, i.e., a jurist, in Toulouse. He devoted himself, however, extensively
and most fruitfully to mathematics so that he may counted as one of
the greatest of mathematicians. Fermat' s name deserves a prominent
place among those of the founders of analytic geometry, of infinitesimal
calculus, and of the theory of probability. Of special significance
however, are his attainments in the theory of numbers. All of his results
in this field appear as marginal notes on his copy of Diophantus, the
famous ancient master of number-theory who lived in Alexandria pro-
bably about 300 A. D., i. e., about 600 years after Euclid. In this form
they were published by his son five years after Fermat' s death. Fermat
himself had published nothing, but he had, by means of voluminous
correspondance with the most significant of his contemporaries, made
his discoveries known, although only in part. It was in that edition
of Diphantus that the famous theorem with which we are now concerned
was found. Fermat wrote concerning it that "he had found a really
wonderful proof, but the margin was too narrow to accommodate it" 1 .
To this day, no one has succeeded in finding a proof of this theorem!

1 See the edition issued by the Paris Academy: (Euvres de Fermat, vol. I,
p. 291. Paris 1891, and vol. Ill, p. 241. Paris 1896.

The Great Format Theorem.

In order to orient ourselves somewhat as to its purport, let us
inquire, as in the case of n = 2, in the first place about the rational
solutions of the equation:

^ n + r, n = 1 ,

i. e., about the relation of the curve which represents this equation to
the totality of the rational points in the >j-plane. For n = 3 and n = 4
the curves have approximately the appearance indicated in Fig. 10, 11
They contain, at least, the points = o, ?7 1 and | = 1 , v\ = when
7

Fig. 11.

n = 3 , and the points = , >/ = 1 an d f = 1 , fl = when
n = 4. The assertion of Fermat means, now, that these curves, unlike
the circle considered above, thread through the dense set of the rational
points without passing through a single one, except those just noted.
The interest in this theorem rests on the fact that all efforts to find
a complete proof of it have been, thus far, in vain. Among those who
have attempted proof, one should, above all, mention Rummer, who
advanced the problem materially by bringing it into relation with the
theory of algebraic numbers, in particular with the theory of the n-th roots

2ix

of unity (cyclotomic numbers). By using the n-th root of 1 , = e n ,
we can, indeed, separate z tl y n into n linear factors, and we may
write the Fermat equation in the form

*= (z -y)(z- ey) (z - &y) ... (z - ? n ~ l y) .

The problem is therefore reduced to the separation of the n-th power
of the integer % into n linear factors which shall be built up from two
integers z and y and the number e, in the manner indicated. Kummer
developed, for such numbers, theories quite similar to those which have
long been known for the case of ordinary integers, theories, that is,
which depend on the notions of divisibility and factorization. One
speaks, accordingly, of integral algebraic numbers, and here, in particular,
of cyclotomic numbers, because of the relation of the number e to the
division of the circle. Fermat' s theorem is, then, for Kummer, a theorem
on factorization in the domain of algebraic cyclotomic numbers. From this

48 Arithmetic: Concerning Special Properties of Integers.

theory he tried to deduce a proof of the theorem. He succeeded, in fact,
for a very large number of values of n t for example for all values -of 'ft
below 100. Among the larger numbers, however, there appeared ex-
ceptional values for which no proof has been found, either by him or
by the later mathematicians who continued his investigations.

I must content myself with these remarks. You will find particulars
concerning the state of the problem, and concerning Rummer's publica-
tions in the Encyclopedia, Vol. I 2 , p. 714, at the end of the report by
Hilbert, Theorie der Algebraischen Zahlkorper. Hilbert himself is among
those who have continued and extended the investigations of Kummer 1 .

It can indeed hardly be assumed that Fermat' s "wonderful proof"
lay in this direction. For it is not very likely that he could have operated
with algebraic numbers at a time when one was not even certain about
the meaning of the imaginary. At that time, also, the theory of numbers
was quite undeveloped. It received at the hands of Fermat himself far-
reaching stimulation. On the other hand, one cannot assume that a
mathematician of Fermat' s rank made an error in his proof, although
such errors have occurred with the greatest mathematicians. Thus we
must indeed believe that he succeeded in his proof by virtue of an
especially fortunate simple idea. But as we have not the slightest
indication as to the direction in which one could search for that idea,
we shall probably expect a complete proof of Fefmat's theorem only through
systematic extension of Rummer's work.

These questions assumed new signifance when our Gottingen Science
Association offered a prize of 100000 marks for the proof of Fermat* s
theorem. This was a foundation of the mathematician Wolfskehl, who
died in 1906. He had probably been interested all his life in Fermat' s
theorem, and he bequeathed from his large fortune this sum for the
fortunate person who should either establish the truth of the theorem
of Fermat, or by means of a single example, exhibit its untruth 2 . Such
a refutation would, be no simple matter, of course, because the theorem
is already proved for exponents below 100 and one would have to start
one's calculations with very large numbers.

It will be clear, from my foregoing remarks, how difficult the winning
of this prize must seem to the mathematician, who understands the
situation and who knows what efforts have been made by Kummer
and his successors to prove the theorem. But the great public thinks

[ l A summarized account of the elementary investigations about Format's
theorem is given in P. Bachmann, Das Fermatsche Problem. Berlin 191 9-]

2 The detailed conditions governing competition for this prize (long since
become valueless) were published in the Nachrichten d. Ges. d. Wissenschaften
zu Gflttingen, business announcements 1908, p. 103 et seq., and copied into many
other mathematical journals (Sec. e. g. Math. Ann. vol. 66, p. 143; Journal fur
Mathematik, vol. 134, p. 313).

Division of the Circle, 49

otherwise. Since the summer of 1907, when the news of the prize was
published in the papers (without authorization, by the way) we have
received a prodigious heap of alleged "proof s". People of all walks of
life, engineers, schoolteachers, clergymen, one banker, many women,
have shared in these contributions. The common thing about them all
is that they have no idea of the serious mathematical nature, of the problem.
Moreover, they have made no attempt to inform themselves regarding
it, but have trusted to finding the solution by a sudden flash of thought,
with the inevitable result that their work is nonsense. One can see
what absurdities are brought forth if one reads the numerous critical
discussions of such proofs by A. Fleck (who is a practising physician
by profession), Ph. Maennchen, and O. Perron, in Archiv fur Mathematik
und Physik 1 . It is amusing to read these wholesale slaughterings, sad
as it is that they are necessary. I should like to mention one example,
which is related to our treatment of the case x 2 + y 2 = z*. The author
seeks a rational parameter representation for the function x n + y n
= z n (n > 2), and finds the result, long known from the theory of
algebraic functions, that this, unlike the case n = 2, is not possible.
Now this person overlooks the fact that a non-rational function can
very well take on rational values for single
rational values of the argument, and he x ' p ane
therefore believes that he has proved the
Fermat theorem.

With this I close my remarks about
Fermat' s theorem and come to the eighth
point of my list, the problem of the division
of the circle. I shall make use here of opera-
tions with complex numbers, x + iy, as-
suming that they are familiar to you, although Fig. 12.

we shall consider them systematically later

on. The problem is to divide the circle into n equal parts, or to construct
a regular polygon of n sides. We identify the circle with the unit circle
about the origin of the complex #y-plane and take x + iy = 1 as the
first of the n points of division (see Fig. 12), in which n is chosen equal
to five) ; then the n complex numbers belonging to the n vertices :

. . fft . . . n n ,, _ . .,

z = x + iy = cos -- h *sm - = e (k= 0, 1, ... , n 1)

satisfy, according to De Moivre's theorem, the equation:

z n = 1 ,

and with this the problem of the division of the circle is resolved into the
solving of this simple algebraic equation. Since it has the rational root

C 1 Vols. XIV, XV, XVI, XVII, XVIII (1901-19H).]

Klein, Elementary Mathematics. 4

50 Arithmetic: Concerning Special Properties of Integers.

z = 1 , z n \ is divisible by z 1 , and there remains for the n 1
other roots the so called cyclotomic equation

z n-i + z n-2 + ... +Z 2 + z+ f = Q /

an equation of degree n 1 , all of whose coefficients are + 1 .

Since ancient times, interest has centered in the question as to what
regular polygons can be constructed with ruler and compasses. It was
known to the ancients that this construction was possible for the
numbers n = 2 h , 3 , 5 (h an arbitrary integer), and likewise for the com-
posite values n = 2 h 3 5 . Here the problem rested until the end of
the eighteenth century when the young Gauss undertook its solution.
He found the desired construction was possible with ruler and compasses

for all prime numbers of the form p2^ 2 ' + \, but for no others. For
the first values /i = 0, 1, 2, 3, 4 this formula yields, in fact, prime
numbers, namely

3, 5, 17, 257, 65537,

of which the first two cases were already known, while the others were
new. Of these the regular polygon of seventeen sides is especially famous.
The fact that it can be constructed with ruler and compasses was first
established by Gauss. Moreover, it is not known for what values of /u
the above formula yields prime numbers. It has been known, for
example, since Euler's time, that for [JL = 5 the number is composite.
I shall not go farther into details, but rather outline the general con-
ditions, and the significance of this discovery. You will find in Weber-
Wellstein details concerning the regular polygon of seventeen sides.

I should like to call to your attention especially the reprint of Gauss'
diary in the fifty-seventh volume of the Mathematische Annalen (1903)
and in Volume X, 1 (1917) of Gauss' Works. It is a small, insignificant
looking book, which Gauss kept from 1796 on, beginning shortly before
his nineteenth birthday. It was precisely the first entry which had to
do with the possibility of constructing the polygon of seventeen sides
(March 30, 1796); and it was this early important discovery which led
Gauss to decide to devote himself to mathematics. The perusal of this
diary is of the highest interest for every mathematician, since it permits
one, farther on, to follow closely the genesis of Gauss' fundamental
discoveries in the field of number theory, of elliptic functions, etc.

The publication of that first great discovery of Gauss appeared as
a short communication in the "Jenaer Literaturzeitung" of June 1, 1796,
instigated by Gauss' teacher and patron, Hofrat Zimmermann, of Braun-
schweig, and accompanied by a short personal note by the latter 1 . Gauss
published the proof later in his fundamental number-theoretic work,

1 Also reprinted in Mathematische Annalen, vol. 57, p. 6 (1903); and in Gauss'
Works, vol. 10, p. 1 (1917).

Division of the Circle. 51

Disquisitiones Arithmeticae 1 in 1801; here one finds for the first time
the negative part of the theorem, which was lacking in his communica-
tion, that the construction with ruler and compasses is not possible for

prime numbers other than those of the form 2 2 + 1 , e.g., for p = 7. I
shall put before you here an example of this important proof of impossi-
bilitythe more willingly because there is such a lack of understanding
for proofs of this sort by the great public. By means of such proofs of
impossibility modern mathematics has settled an entire series of famous
problems, concerning the solution of which many mathematicians had
striven in vain since ancient times. I shall mention, besides the con-
struction of the polygon of seven sides, only the trisection of an angle
and the quadrature of the circle with ruler and compasses. Nevertheless
there are surprisingly many persons who devote themselves to these
problems without having a glimmering of higher mathematics and
without even knowing or understanding the nature of the proof of
impossibility. According to their knowledge, which is mostly limited
to elementary geometry, they make trials, by drawing, as a rule, auxiliary
lines and circles, and multiply these finally in such number that no
human being, without undue expenditure of time, can find his way out
of the maze and show the author the error in his construction. A
reference to the arithmetic proof of impossibility avails little with such
persons, since they are amenable, at best, only to a direct consideration
of their own "proof " and a direct demonstration of its falsity. Every
year brings to every even moderately known mathematician a heap of
such consignments, and you also, when you are at your posts, will get
such proofs. It is well for you to be prepared in advance for such ex-
periences and to know how to hold your ground. Perhaps it will be well
for you, then, if you are master of a definite proof of impossibility in
its simplest form.

Accordingly, I should like to give you, in detail, the proof that it
is impossible to construct the heptagon with ruler and compasses in the
sense of geometry of precision. It is well known that every construction
with ruler and compasses finds its arithmetic equivalent in a succession
of square roots, placed one above another, and, conversely, that one
can represent geometrically every such square root by the intersection
of lines and circles. This you can easily verify for yourselves. We can
formulate our assertion analytically, then, by saying that the equation
of degree six

2 6 + * 5 + 2 4 + 2 3 + Z 2 + Z + \ = ,

which characterizes the regular heptagon, cannot be solved by a succession
of square roots in finite number. Now this is a so-called reciprocal equation,

1 Reprinted Works, vol. I.

J2 Arithmetic: Concerning Special Properties of Integers.

i. e., it has, for every root z t also \\z as a root. This becomes obvious
if we write it in the form :

(1) * 3 + * 2 + * + 1 + y + ? + 73 = 0.

We can reduce by half the degree of such an equation, if we take

as a new unknown. By easy calculation, we obtain for x the cubic
equation

(2) x 3 + x 2 2x \ = 0,

and one sees at once that the equations (1) and (2) are, or are not, both
solvable by square roots. Moreover, we can represent x geometrically
in connection with the construction of the heptagon. For, if we consider

the unit circle in the complex plane, we see easily that the following re-

/^ _

lations are obvious. If one designates by (? = the central angle of
the regular heptagon, and remembers that z = cos q> + i sin <p and

= cos <p i sin <p are the two vertices of the heptagon nearest to

% = \ , then x = z + = 2 cos <p (Fig. 13). Thus, if one knows x, one

can at once construct the heptagon.

We must now show that the cubic equation (2) cannot be solved by

square roots. The proof falls into an arithmetic and an algebraic part.

z . p l ane We shall start by showing that the equa-

tion (2) is irreducible, i. e. that its left side
cannot be separated into two factors
whose coefficients are rational numbers.
Let us assume that the equation is re-
ducible. Then its left side must have a
linear factor with rational coefficients,
and hence it must vanish for a rational
Fjg 13 number pfq t where p and q are integers

without a common divisor. But that

means that /> 3 + p 2 q 2pq 2 q* =s= 0, or that p*, and therefore p
itself, is divisible by q. In the same way it follows that <? 3 , and hence q,
must be divisible by p . Consequently p = ? and the equation (2) must
have the root x = 1 . But inspection shows that this is not the case.
The second part of the proof consists, in showing that an irreducible
cubic equation with rational coefficients is not solvable by square roots. It
is essentially algebraic in nature, but because of the connection I shall
give it here. Let us make the assertion in positive form. If a cubic
equation with rational coefficients A , B , C :

(8) /(*) = x* + Ax*

Division of the Circle.

can be solved by square roots, it must have a rational root, i. e., it is reducible.
For the existence of a rational root # is equivalent to the existence of
a rational factor % oc of / (x) and thus to reducibility. It is most
important that this proof be preceded by a classification of all expressions
that can be built up with square roots, or, more precisely, of all expressions
that can be built up with square roots and rational numbers, in finite number,
by means of rational operations. A concrete example of such a number is

where a, b, . . ., f are rational numbers. Of course we are talking only
about square roots which cannot be extracted rationally. All others must
be simplified. Every such expression is a rational function of a certain
number of square roots. In our example there are three. We shall first
consider a single such square root, whose radicand, however, may have
a form as complicated as one pleases. By its "order" we shall understand
the largest number of root signs which appear in it, one above another. In
the preceding example, oc , the roots of the numerator have the orders
2 and 1 , respectively, while that of the denominator has the order 3 -
In the case of a general square root expression we examine the orders
of the different "simple square root expressions" of the sort just discussed,
out of which the general expression is rationally constructed, and we
designate the largest among them as the order fi of the expression in question.
In our example, /* = 3. Now several "simple square root expressions 1 '
of order jm might appear in our expression and we consider their number,
n, the "number of terms" of order p, as a second characteristic. This
number is thought of as so determined that no one of the n simple
expressions of order p can be rationally expressed in terms of the others of
order ft, or of lower order. For example, the expression of order 1

y 2 + y~3 + y~6

has 2, not 3 , as the "number of terms" since /6 = }2 V3 . The example
oi given above has n = 1 .

We have thus assigned to every square root expression two finite
numbers // , n which we combine in the symbol (/n , n) as the "characteristic"
or "rank" of the root expression. When two root expressions have different
orders we assign a lower rank to the one of lower order', when the orders
are the same, the lower number of terms determines the lower rank.

Now let us suppose that a root x l of the cubic equation (8) is expres-
sible by means of square roots; and, to be explicit, by means of an
expression of rank (p,ri). Selecting one of the n terms ]/ R of rank fi , let x
be written in the form

54 Arithmetic: Concerning Special Properties of Integers.

where &, 0, y, d contain at most n \ terms of order /A and where R
is of order p \ . Here y d ]/R is certainly different from zero ;
for y 8 y^R = would imply either 6 = 7 = 0, which is obviously
impossible, or]TR=y : 6, i.e., /7? would be rationally expressible by
means of the other (n 1) terms of order ,a, which appear in x, and
hence it would be superfluous. Multiplying numerator and denominator
by y (>1/R, we find

,--

r - L + v V K >

where P, Q are rational functions of <x,fi,y,d, that is, they contain
at most (n 1) terms of order //, and, besides, only those of order
// 1 , so that they have at most the rank (ft, n 1). Substituting this
value of x in (8), we get

f(x l ] = (P + QfRf + A(P + <?]/tf) 2 + B(P + QfR) + C = 0,
and when we remove parentheses we obtain a relation of the form

where M, 2V are polynomials in P, Q, R, that is, rational functions of

a, ft, y> 6, R.'IfN^ 0, we should have ]//? == -M/N, i.e., ]//?" would
be expressible rationally in terms of #,/?, y, 6, 7?, that is, by means of
the other (n 1) terms of order // and others of lower order. But
that is impossible, as remarked above, according to the hypothesis.
Thus it follows necessarily that N = and hence also M = . From
this we may conclude, that

** = P- QfR

is also a root of the cubic equation (8). For a comparison with the last
equations yields at once

/(*,) =M - NfR = Q.

The proof may now be finished very simply and surprisingly. If x. 3 is
the third root of our cubic equation, we have

*! + # 2 + #3 = A ,

and hence x 3 = A (x l + x 2 ) = A 2 P

is of the same rank as P and therefore certainly of lower rank than x l .
If x 3 is itself rational, our theorem is proved. If not, we can make
it the starting point of the same series of deductions. It appears that,
in the case of the other roots, the higher rank must have been an illusion,
so that, in particular, one of them has, actually, lower rank than x 3 . If
we keep this up, back and forth among the roots, we see, each time,
that the rank is really lower than we had thought. We must, then,
of necessity, come finally to a root with the order ^ = . This demon-

Ordinary Complex Numbers. 55

strates the existence of a rational root of the cubic equation. We cannot
continue our procedure beyond this point. The two other roots must

then be, either themselves rational, or else of the form P = Q]/R,
where P, Q , R are rational numbers. Hence we have shown that f (x)
separates into a quadratic and a linear rational factor and is therefore
reducible. Every irreducible cubic equation, and in particular, our equation
for the regular heptagon, is insoluble by means of square roots. The proof
is therefore complete that the regular heptagon cannot be constructed with
ruler and compasses.

You observe how simply and obviously this proof proceeds, and
how little knowledge it really presupposes. For all that, some of the
steps, especially the explanation of the classification of square root ex-
pressions, demand a certain measure of mathematical abstraction.
Whether the proof is simple enough to convince one of those mathe-
matical laymen, mentioned above, of the futility of his attemps at an
elementary geometric proof, I do not presume to decide. Nevertheless
one should try to explain the proof slowly and clearly to such a person.

In conclusion, I shall mention some of the literature on the question
of regular polygons together with some, on the broader question of
geometric constructibility in general which we have touched upon on
this occasion. First of all, there is again Weber-Wellstein I (Sections 1 7
and 18 in the fourth edition). Next let me mention the souvenir booklet
Vortrdge iiber ausgewdhlte Fragen der Elementargeometrie 1 * which I pre-
pared in 1895, on the occasion of a gathering of teachers in Gottingen.
I might mention, as a more detailed and comprehensive substitute for
this little book (which is out of print) the German translation, Fragen
der Elementargeometrie***, of a compilation by F. Enriques in Bologna,
where you will find information on all allied questions.

I leave now the discussion of number theory, reserving the last
point, the transcendence of n , for the conclusion of this course of lectures,
and turn, in the next chapter, to our final extension of the number system.

IV. Complex Numbers.
1. Ordinary Complex Numbers

Let me give, as a preliminary, some historical facts. Imaginary
numbers are said to have been used first, incidentally, to be sure, by
Cardan in 1 545, in his solution of the cubic equation. As for the further

1 Worked up by F. Tagert. Leipzig 1895-

2 Teilll: Die geometrischen Aufgaben, ihre Losung und Losbarkeit. Deutsch
von H. Fleischer. Leipzig 190?. [2. Aufl. 1923-] See also Young, J. W. A. t Mono-
graphs on Topics in Modern Mathematics.

* Translation by Beman and Smith: Famous Problem of Geometry. Ginn,
reprinted by Stechert, New York.
** Problems of Elementary Geometry.

56 Arithmetic: Complex Numbers.

development, we can make the same statement as in the case of negative
numbers, that imaginary numbers made their own way into arithmetic
calculation without the approval, and even against the desires of individual
mathematicians, and obtained wider circulation only gradually and to
the extent to which they showed themselves useful. Meanwhile the mathe-
maticians were not altogether happy about it. Imaginary numbers
long retained a somewhat mystic coloring, just as they have today for

every pupil who hears for the first time about that remarkable i = V 1 .
As evidence, I mention a very significant utterance by Leibniz in the
year 1702, "Imaginary numbers are a fine and wonderful refuge of the
divine spirit, almost an amphibian between being and non-being". In
the eighteenth century, the notion involved was indeed by no means
cleared up, although Euler, above all, recognized their fundamental
significance for the theory of functions. In 1748 Euler set up that remark-
able relation:

e ix = cos# + isinx

by means of which one recognizes the fundamental relationship among
the kinds of functions which appear in elementary analysis. The
nineteenth century finally brought the clear understanding of the nature
of complex numbers. In the first place, we must emphasize here the
geometric interpretation to which various investigators were led about
the end of the century. It will suffice if I mention the man who certainly
went deepest into the essence of the thing and who exercised the most
lasting influence upon the public, namely Gauss. As his diary, men-
tioned above, proves incontrovertibly, he was, in 1 797, already in full
possession of that interpretation, although, to be sure, it was published
very much later. The second achievement of the nineteenth century
is the creation of a purely formal foundation for complex numbers,
which reduces them to dependence upon real numbers. This originated
with English mathematicians of the thirties, the details of which I
shall omit here, but which you will find in Hankel's book, mentioned
above.

Let me now explain these two prevailing foundation methods. We
shall take first the purely formal standpoint, from which the consistency
of the rules of operation among themselves, rather than the meaning
of the objects, guarantees the correctness of the concepts. According
to this view, complex numbers are introduced in the following manner,
which precludes every trace of the mysterious.

1 . The complex number x + iy is the combination of two real numbers
x,y, that is, a number-pair, concerning which one adopts the conven-
tions which follow.

2. Two complex numbers x + iy, x' + iy f are called equal when
x = x' ,y = y'.

Ordinary Complex Numbers. 57

3. Addition and subtraction are defined by the relation

(* + iy} (*' + iy'} = (* *'} + i(y y') .

All the rules of addition follow from this, as is easily verified. The mono-
tonic law alone loses its validity in its original form, since complex
numbers, by their nature, do not have the same simple order in which
natural or real numbers appear by virtue of their magnitude. For the
sake of brevity I shall not discuss the modified form which this gives
to the monotonic law.

4. We stipulate that in multiplication one operates as with ordinary
letters, except that one always puts i 2 = 1; in particular, that

(x + iy) (x f + iy') = (**' - yy') + i(xy' + x'y).

It is easy to see that, with this, all the laws of multiplication hold, with
the exception of the monotonic law, which does not enter into consideration.

5. Division is defined as the inverse of multiplication] in particular,
we may easily verify that

1 = _5_ __ j y

x + iy x 2 + y 2 x 2 + y 2 '

This number always exists except for x y = 0, i.e., division by zero
has the same exceptional place here as in the domain of real numbers.

It follows from this that operations with complex numbers cannot
lead to contradictions, since they depend exclusively upon real numbers
and known operations with them. We shall
assume here that these are devoid of contra-
diction.

Besides this purely formal treatment, we
should of course like to have a geometric, or
otherwise visual, interpretation of complex
numbers and of operations with them, in which
we might see a graphical foundation of consi-
stency. This is supplied by common geometric

interpretation, which, as you all know and as

we have already mentioned, looks upon the
totality of points (x,y) of the plane in an pig. u.

xy-coordinate system as representing the totality

of complex numbers z = x -\- iy. The sum of two numbers z, a follows
by means of the familiar parallelogram construction with the two
corresponding points and the origin 0, while the product z a is
obtained by constructing on the segment 02 a triangle similar to
001, where 1 is the point (x = 1 , y = 0) (Fig. 14). In brief, addition
z 1 == z + a is represented by a translation of the plane into itself, mul-
tiplication z 1 = z ' a by a similarity transformation, i.e., by a turning
and a stretching, the origin remaining fixed. From the order of the points

Jg Arithmetic: Complex Numbers.

in the plane, considered as representatives of complex numbers, one
sees at once what takes the place here of the monotonic laws for real
numbers. These suggestions will suffice, I hope, to recall the subject
clearly to your memory.

I must call to your attention the place in Gauss in which this founda-
tion of complex numbers, by means of their geometric interpretation,
is set out with full emphasis, since it was this which first exhibited the
general importance of complex numbers. In the year 1831 Gauss'
researches carried him into the theory especially of integral complex
numbers a + ib, where a, b are real integers, in which he developed
for the new numbers the theorems of ordinary number theory concerning
prime factors, quadratic and biquadratic residues, etc. We mentioned
such generalizations of number theory, in connection with our discussion
of Fermat's theorem. In his own abstract 1 of this paper Gauss
expresses himself concerning what he calls the "true metaphysics of
imaginary numbers". For him, the right to operate with complex
numbers is justified by the geometric interpretation which one gives
to them and to the operations with them. Thus he takes by no means
the formal standpoint. Moreover, these long, beautifully written ex-
positions of Gauss are extremely well worth reading. I mention here,
also, that Gauss proposes the clearer word "complex", instead of
"imaginary", a name that has, in fact, been adopted.

2. Higher Complex Numbers, especially Quaternions

It has occurred to everyone who has worked seriously with complex
numbers to ask if we cannot set up other, higher, complex numbers,
with more ne wunits than the one i and if we cannot operate with them
logically. Positive results in this direction were obtained about 1840
by H. Grassmann, in Stettin, and W. R. Hamilton, in Dublin, indepen-
dently of each other. We shall examine the invention of Hamilton, the
calculus of quaternions, somewhat carefully later on. For the present
let us look at the general problem.

We can look upon the ordinary complex number x + iy as a linear

combination

% 1 + y i

formed from two different "units" \ and i , by means of the real parameters
x and y . Similarly, let us now imagine an arbitrary number, n , of units
e , e 2 , . . . , e n all different from one another, and let us call the totatily
of combinations of the form x = x& + x 2 e 2 + . . . , + x n e n a higher
complex number system formed from them with n arbitrary real numbers
x lt x 2 , . . ., x n . If there are given two such numbers, say x, defined
above, and

y =

1 See Werke, vol. II.

Higher Complex Numbers, especially Quaternions. 59

it is nearly obvious that we should call them equal when, and only when,
the coefficients of the individual units, the so called "components" of the
number, are equal in pairs

The definition of addition and siibtr action, which reduces these operations
simply to the addition and subtraction of the components,

* it y = (*i yi)*i + (* a y 8 )*2 + ...,+ (x n y n ) e n ,

is equally obvious.

The matter is more difficult and more interesting in the case of
multiplication. To start with, we shall proceed according to the general
rule for multiplying letters, i.e., multiply each i-th term of x by every
k-ih term of y (i , k = \ , 2 , . . . , n) . This gives :

x-y= ^ *y**i**-

(', *=1, .... n)

In order that this expression should be a number in our system, one must
have a rule which represents the products d e* as complex numbers
of the system, i. e., as linear combinations of the units. Thus one must
have n 2 equations of the form:

Then we may say that the number

*-y = Z I Z

(1=1. ..., n)l(i, /--I, ..., n)

will always belong to our complex number system. Each particular
complex number system is characterized by the method of determining
this rule for multiplication, i.e., by the table of the coefficients Cud.

If one now defines division as the operation inverse to multiplication,
it turns out that, under this general arrangement, division is not always
uniquely possible, even when the divisor does not vanish. For, the
determination of y from x y = z requires the solution of the n linear
equations xiytCiu = Zi for the n unknowns y lf . . ., y n , and these

would have either no solution, or infinitely many solutions, if their
determinant happened to vanish. Moreover, all the zi may be zero
even when not all the Xi or not all the y^ vanish, i.e., the product of two
numbers can vanish without either factor being zero. It is only by a skillful
special choice of the numbers dki that one can bring about accord here
with the behavior of ordinary numbers. To be sure, a closer investigation
shows, when n > 2 , that, to attain this, we must sacrifice one of the other
rules of operation. We choose as the rule that fails to be satisfied, one
which appears less important under the circumstances.

60 Arithmetic: Complex Numbers.

Let us now follow up these general explanations by a more detailed
discussion of quaternions as the example which, by reason of its applica-
tions in physics and mathematics, constitutes the most important higher
complex number system. As the name indicates, these are four-term
numbers (n = 4) ; as a sub-class, they include the three-term vectors,
which are generally known today, and which are sometimes discussed
in the schools.

As the first of the four units with which we shall construct quaternions,
we shall select the real unit \ , (as in the case of ordinary complex num-
bers). We ordinarily denote the other three units, as did Hamilton,
by i,j,k, so that the general from of the quaternion is

p = d + ia + jb + kc,

where a,b,c,d are real parameters, the coefficients of the quaternion.
We call the first component, the one which is multiplied by 1, and
which corresponds to the real part of the common complex number,
the "scalar part" of the quaternion, the aggregate ai + bj + ck of the
other three terms its "vector part" .

The addition of quaternions follows from the preceding general
remarks. I shall give an obvious geometric interpretation, which goes
back to that interpretation of vectors which is familiar to you. We
imagine the segment, corresponding to the vector part of p , and having
the projections a,b,c on the coordinate axes, as loaded with a weight
equal to the scalar part. Then addition of p and p' = d' + ia' + jb' + kc'
is accomplished by constructing the resultant of the
two segments, according to the well known parallelogram
law of vector addition (see Fig. 15), and then loading it
w jth the sum of the weights, for this would then in fact
represent the quaternion:

(1) 'p + p'=(d + d') + i(a + a')+j(b + b')+k(c + c').

We come first to specific properties of quaternions
when we turn to multiplication. As we saw in the general
Fig. 15. case, these properties must be implicit in the conventions

adopted as to the, products of the units. To begin with,
I shall indicate the quaternions to which Hamilton equated the
sixteen products of two units each. As its symbol indicates, we shall
operate with the first unit 1 as with the real number 1, so that:

(2a) l a = l f i-l =! = , / 1 =!/ = /, -1=1 -k = k.

As something essentially new, however, we agree that, for the squares of

the other units:

(2b) i a = / 2 = *=-!,

Higher Complex Numbers, especially Quaternions. fi\

and for their binary products:

(2c) jk = +i, ki = j, ij = +k

whereas for the inverted position of the factors:
(2d) kj = i t ik=j, ji = k.

One is struck here by the fact that the commutative law for multiplication
is not obeyed. This is the inconvenience in quaternions which one must
accept in order to rescue the uniqueness of division, as well as the theorem
that a product should vanish only when one of the factors vanishes.
We shall show at once that not only this theorem but also all the other laws
of addition and multiplication remain valid, with this one exception, in
other words, that these simple agreements are very expedient.
We construct, first, the product of two general quaternions

p = d + ia + jb + kc and q = w + ix + jy + kz.
Let us start from the equation

q' = p - q = (d -\- ia + j b + kc) (w + ix + jy + kz) ;

and let us multiply out term by term. In carrying out this multiplication,
we must note the order in the case of the units i,j , k. We must follow
the commutative law for products composed of the components a,b,c,d,
and for products of components and one unit, we must replace the
products of units in accordance with our multiplication table, and we
must then collect the terms having the same unit. We must then
collect the terms having the same unit. We then have

q' = pq = w' + ix' + jy' + kz' = (dw ax by cz)

+ i(aw + dx + bz cy)

(3)

+ j(bw + dy + ex az)

+ k (cw + dz + ay bx) .

The components of the product quaternion are thus definite simple
bilinear combinations of the components of the two factors. If we
invert the order of the factors, the six underscored terms change their
signs, so that q' p, in general, is different from p q, and the difference
is more than a change of sign as was the case with the individual units.
Although the commutative law fails for multiplication, the distribu-
tive and associative laws hold without change. For, if we construct on
the one hand p(q + q^ , on the other pq + pq l by multiplying out
formally without replacing the products of the units, we must, of
necessity, get identical results, and no change can be brought about
by then using the multiplication table. Further, the associative law
must hold in general, if it holds for the multiplication of the units.

Arithmetic: Complex Numbers.

But this follows at once from the multiplication table, as the following
example shows:

In fact, we have:
and

(*;)* =

i(jk) = i i == 1.

We shall now take up division. It will suffice to show that for every
quaternion p ~ d -\- ia -}- jb -\- kc there is a definite second one, q, such
that:

We shall denote q appropriately by \jp . Division in general can be
reduced easily to this special case, as we shall show later. In order to
determine q, let us put, in equation (3),

q 9 ^ \ = \ + Q'i + o-j + 0- k,

and obtain, by equating components, the following four equations for
four unknown components x,y,z,w of q:

dw ax by cz = \
aw + dx cy + bz =
bw -\- ex + dy az =
cw b% + ay + dz 0.

The solvability of such a system of equations depends, as is well known,
upon its determinant, which, in the case before us, is a skew symmetric
determinant, in which all the elements of the principal diagonal are the
same, and all the pairs of elements which are symmetrically placed with
respect to that diagonal are equal and opposite in sign. According to
the theory of determinants, such determinants are easily calculated;
and we find

d a

-b

a d

b c

c -b

By direct calculation this result can be easily verified. The real elegance
of Hamilton's conventions depends upon this result, that the determinant
is a power of the sum of squares of the four components of p\ for it
follows that the determinant is always different from zero except when
a = b = c = d = 0. With this one self evident exception (p = 0),
the equations are uniquely solvable and the reciprocal quaternion q is
uniquely determined..

Higher Complex Numbers, especially Quaternions. gi

The quantity

T = ]/0 2 + b 2 + c 2 + d 2

plays an important role in the theory, and is called the tensor of p.
It is easy to show that these unique solutions are

a b c d

x ~Y 2 , y -y^ z ~~"Y2' w = ~jv

so that we have as the final result

1 1 d ia jb he

~~ ~~"

If we introduce the conjugate value of p , as in ordinary complex numbers :

p =. d ia jb kc,
we can write the last formula in the form

1= i

p T*
or

p.p = T 2 - a 2 + b 2 + c 2 + d 2 .

These formulas which are immediate generalizations of certain properties
of ordinary complex numbers. Since p is also the number conjugate
to p, it follows also that:

so that the commutative law holds in this special case.

The general problem of division can now be solved. For, from the
equation

it follows, by multiplication by !//>, that

-*--.

whereas the equation

9 ' P = l'>
which one gets by changing the order of the factors, has the solution

This solution is different, in general, from the other.

Now we must inquire whether there is a geometric interpretation of
quaternions in which these operations, together with their laws, appear
in a natural form. In order to arrive at it, we start with the special
case in which both factors reduce to simple vector s t i.e., in which the

Arithmetic: Complex Numbers.

scalar parts w,d, are zero,
becomes

The formula (3) for multiplication then

(<L t b,c)

q 1 = p q = (ia + jb + kc) (ix + jy + kz)

= (ax + by.+ cz) + i(bz cy) + j(cx az) + k(ay bx),

i. e., when each of two quaternions reduces to a vector, their product consists
of a scalar and a vector part. We can easily bring these two parts into
relation with the different kinds of vector multiplication which are in
use. The notions of vector calculus, which is far more wide spread than
quaternion calculus, go back to Grassmann, although
the word vector is of English origin. The two kinds
of vector product with which one usually operates
are designated now, mostly, by inner (scalar) product
ax + by + cz (i.e., the scalar part of the above
quaternion product, except for the sign), and outer
(vector) product i (bz cy)+j (ex az) + k (ay bx) ,
(i.e., the vector part of the quaternion product. We shall give a geo-
metric interpretation of each part separately.

Let us lay off both vectors (a , b , c) and (x , y , z) , as segments, from
the origin (Fig. 16). They terminate in the points (a>b,c) and (x,y t z)

respectively, and have the lengths l = ^a 2 + b* + c 2 and /' = }/ x 2 + y * + z 2 .

If (p is the angle between these two segments, then, according to well

known formulas of analytic geometry,
which I do not need to develop here,
the inner product is:

ax + by + cz = I

Fig. 16.

cos<p;

and the outer product, on the other
hand, is itself a vector, which, as is
easily seen, is perpendicular to the
plane of I and I' and has the length
I I' sin??.

It is essential now to decide as to
the sense of the product vector, i.e.,
toward which side of the plane deter-
mined by I and V one is to lay off
this vector. This sense is different
according to the coordinate system

which one chooses. As you know, one can choose two rectangular co-
ordinate systems which are not congruent , i.e., which cannot be made to
coincide with one another, by holding, say, the y- and the 2-axis fixed
and reversing the sense of the #-axis. These systems are then sym-
metric to each other, like the right and the left hand (Fig. 1 7) . The distinction
between them can be borne in mind by the following rule: In the one

Fig. 17.

Quaternion Multiplication Rotation and Expansion. 55

system, the x, y, and z axis lie like the outstretched thumb, fore finger and

middle finger, respectively, of the right hand] in the other, like the same

fingers of the left hand. These two systems are used confusedly in the

literature; different habits obtain in different countries, in different

fields, and, finally, with different writers, or even with the same writer.

Let us now examine the simplest case, where p = i, q = j , these being

the unit lengths laid off on the x and y axis. Then, since '/ = ,

the outer vector product is the unit length laid off

on the 2-axis. (See Fig. 18.) Now one can trans- \

form i and j continuously into two arbitrary vectors

p and q so that k transforms continuously into the

vector component of p q without going through

zero. Consequently the first factor, the second factor,

and the vector product must always lie, with respect to Fig. is.

each other, like the x, y, and z-axis of the system of

coordinates, i.e., right-handed (as in Fig. 18) or left-handed (as in Fig. 16),

according to the choice of coordinate system. (In Germany, now, the choice

indicated in Fig. 18 is customary.)

I should like to add a few words concerning the much disputed
question of notation in vector analysis. There are, namely, a great many
different symbols used for each of the vector operations, and it has been
impossible, thus far, to bring about a generally accepted notation.
At the meeting of natural scientists at Kassel (1903) a commission was
set up for this purpose. Its members, however, were not able even to
come to a complete understanding among themselves. Since their
intentions were good, however, each member was willing to meet the
others part way, so that the only result was that about three new
notations came into existence! My experience in such things inclines
me to the belief that real agreement could be brought about only if
important material interests stood behind it. It was only after such
pressure that, in 1881, the uniform system of measures according to
volts, amperes, and ohms was generally adopted in electrotechnics and
afterward settled by public legislation, due to the fact that industry
was in urgent need of such uniformity as a basis for all of its calculations.
But there are no such strong material interests behind vector calculus,
as yet, and hence one must agree, for better or worse, to let every
mathematician cling to the notation which he finds the most convenient,
or if he is dogmatically inclined the only correct one.

3. Quaternion Multiplication Rotation and Expansion

Before we proceed to the consideration of the geometric meaning
of multiplication of general quaternions, let us consider the following
question. Let us consider the product q' = p q of two quaternions p
and q, and let us replace p and q by their conjugates p and q, that

Klein, Elementary Mathematics. 5

66 Arithmetic: Complex Numbers.

is, let us change the signs of a,b,c,x,y,z. Then the scalar part of
the product, as given in (3), p. 61, remains unchanged, and only those
factors of i,j,k which are not underscored will change sign. On the
other hand, if we also reverse the order of the factors p and q, the
factors of i,j,k which are underscored will change sign. Hence the
product q'=q'p is precisely the conjugate of the original product q*\
and we have

q' = p-q, q' = q-p,

where q' is the conjugate of q'. If we multiply these two equations
together, we obtain

In this equation the order of the factors is essential, since the com-
mutative law does not hold. We may apply the associative law, however,
and we may write

q'-q' = p'(q-q) -p.

Since we have, by .p. 63,
we may write

q q = x * + 3/2

y 2

The middle factor on the right is a scalar, and the commutative law
does hold for multiplication of a scalar by a quaternion, since M p
= Md + i(Ma) + j(MV) + k(Mc) = pM . Hence we have

w'* + x'* + y' 2 + z'* = pp(w 2 + x* + y 2 + z 2 ),
and, since p -p is the square of the tensor of p, we find 1
(I) w' z + x'* + y' 2 + *' 2 = (d* + a* + b* + c 2 ) (w* + x* + y 2 + * 2 ),

that is, the tensor of the product of two quaternions is equal to the product
of the tensors of the factors. This formula can be obtained also by direct
calculation, by taking the values of w', %' , y', z' from the formula for
a product given on p. 61 .

We shall now represent a quaternion as the segment joining the
origin of a four-dimensional space to the point (x , y , z , w) , in a manner
exactly analogous to the representation of a vector in three-dimensional
space. It is no longer necessary to apologize for making use of four-
dimensional space, as was the custom when I was a student. All of
you are fully aware that no metaphysical meaning is intended, and that
higher dimensional space is nothing more than a convenient mathematical
expression which permits us to use terminology analogous to that of

1 This formula, in all that is essential, occurs in Lagrange's works.

Quaternion Multiplication Rotation and Expansion. 67

actual space representation. If we regard p as a constant, that is, if
we regard a,b,c,d as constants, the quaternion equation

9' = P ' q

represents a certain linear tranformation of the points (x t y,z,w) of
the four-dimensional space into the points (x f , y' ', 2', w'), since the
equation assigns to every four-dimensional vector q another vector q'
linearly. The explicit equations for this transformation, i.e., the ex-
pressions for x' 9 y',z', w' as linear functions oix,y,z,w, may be obtained
by comparison of the coefficients of the product formula (3), p. 61.
The tensor equation (I) shows that the distance of any point from the
origin, ^x z + y 2 + z 2 + w 2 , is multiplied by the same constant factor
T = l/> + & 2 + 'c*~+d*, for all points of the space. Finally, by
p. 62, the determinant of the linear transformation is surely positive.

It is shown in analytic geometry of three-dimensional space that
if a linear transformation of the coordinates x,y,z is orthogonal (that
is, if it carries the expression % 2 + y 2 + z 2 into itself), and if the deter-
minant of the transformation is positive, the transformation represents
a rotation about the origin. Conversely, any rotation can be obtained in
this manner. If the linear transformation carries x 2 + y 2 + z z into
the similar expression in x', y', z' multiplied by a constant factor T 2 ,
however, and if the determinant is positive, the transformation re-
presents a rotation about the origin combined with an expansion in the
ratio T about the origin, or, briefly, a rotation and expansion.

The facts just mentioned for three-dimensional space may be ex-
tended to four-dimensional space. We shall say that our transformation
of four-dimensional space represents in precisely the same sense a
rotation and expansion about the origin. It is easy to see, however, that
in this case we do not obtain the most general rotation and expansion
about the origin. For our transformation contains only four arbitrary
constants, namely, the components a , b , c , d of p , whereas, as we shall
show immediately, the most general rotation and expansion about the
origin in the four-dimensional space K 4 contains seven arbitrary con-
stants. Indeed, in order that the general linear transformation should
be a rotation and expansion, we must have

If we replace x',y',z',w' by linear integral functions of x,y,z>w t
we obtain a quadratic form in four variables, which contains (4 5)/2 = 10
terms. Equating coefficients, we obtain ten equations. Since T is still
arbitrary, these reduce to nine equations among the sixteen coefficients
of the transformation. Hence there remain seven arbitrary constants.
It is remarkable that in spite of this the most general rotation and
expansion can be obtained by quaternion multiplication. Let n = t> + i&

68 Arithmetic: Complex Numbers.

+ j'P + ky be another constant quaternion. Then we may show, just as
before, that the transformation q' = q-n, which differs from the
preceding one only in that the order is reversed, represents a rotation
and expansion of jR 4 . Hence the combined transformation

(II) q' = p-q-7i = (d + ia + jb + kc)-q-(d + ioc + jfi + ky)

also represents such a rotation and expansion. This transformation
contains only seven (not eight) arbitrary constants, for the trans-
formation remains unchanged if we multiply a , b , c , d by any real
number and divide & , ft , y , <5 by the same number. It is therefore
plausible that this combined transformation represents the general
rotation and expansion of four-dimensional space. This beautiful result
is actually true, as was shown by Cayley. I shall restrict myself to the
mention of the historical fact, in order not to be drawn into too great
detail. The formula is given in Cay ley's paper on the homographic
transformation of a surface of the second order into itself 1 , in 1854, and also
in certain other papers of his 2 .

This formula of Cayley' s has the great advantage that it enables
us to grasp at once the combination of two rotations and expansions.
Thus, if a second rotation and expansion be given by the equation

q" = w" + ix" + jy" + kz" = p f -q' n' ,
where p' and n' are new given quaternions, we find, by (II),

?" = P' ' (P <1 ' ?*) ri ,
whence, by the associative law,

<f=(P'. P ).q.(n.n'}
or

q" = r q Q

where r = p' p and Q = n n' are definite new quaternions. We have
therefore obtained an expression for the rotation and expansion that
carries q into q" in precisely the old form, and we see that the multipliers
which precede and follow q in the quaternion product are, respectively,
the products of the corresponding multipliers of q in the separate trans-
formations which were combined, the order of the factors being neces-
sarily as shown in the formula.

This four-dimensional representation may seem unsatisfactory, and
there may be a desire for something more tangible which can be re-
presented in ordinary three-dimensional space. We shall therefore
show that we can obtain similar formulas for the similar three-dimensional

1 Journal fur Mathematik, 185$. Reprinted in Cayley's Collected Papers, vol. 2,
p. 133. Cambridge 1889.

2 See, for example, Recherches ultMeures sur les determinants gauches, loc. cit.,
p. 214.

Quaternion Multiplication Rotation and Expansion. 69

operations by a simple specialization of the formulas just given. Indeed
the importance of quaternion multiplication for ordinary physics and
mechanics is based upon these very formulas. I have said "ordinary",
because I do not desire at this point to explain those generalizations
of these science for which the preceding formulas apply without any
modification. These generalizations are more immediate, however, than
you may suppose. The new developments of electrodynamics which
are associated with the principle of relativity, are essentially nothing
else than the logical use of rotations and expansions in a four-dimensional
space. These ideas have been presented and enlarged upon recently
by Minkowski 1 .

Let us remain, however, in three-dimensional space. In such a space,
a rotation and expansion carries a point (x, y, z) into a point (x f ', y r , z'}
in such a way that

*'2 + y'2 + j'2 = ^2(^2 + y2 + ^2) ^

where M denotes the ratio of expansion of every length. Since the
general linear transformation of (x 9 y 9 z) into (x' 9 y' 9 z') contains nine
coefficients, and since the left-hand side of the preceding equation,
after the insertion of the values of x', y', z'> becomes a quadratic form
in x , y , z with six terms, the comparison of coefficients in the preceding
equation leads to six equations, which reduce to five if the value
of M is supposed arbitrary. Therefore the nine original coefficients
of the linear transformation, which are subject to these five conditions,
are reduced to four arbitrary constants. (Compare p. 67.) If such a
linear transformation has a positive determinant, it represents, as was
stated on p. 67, a rotation of space about the origin, together with an
expansion in the ratio 1/M. If the determinant is negative, however,
the transformation represents a rotation and expansion, combined with
a reflection, such as, for example, the reflection defined by the equations
x = x', y = y', z = z'. Moreover, it can be shown that the deter-
minant of the transformation must have one of the two values M 3 .
In order to represent these relationships by means of quaternions,
let us first reduce the variable quaternions q and q' to their vectorial
parts :

q' = ix' -f. jy' -f kz', q = ix + jy + kz,

which we shall think of as the three-dimensional vectors joining the
origin to the positions of the point before and after the transformation,
respectively. We shall show that the general rotation and expansion

1 Since this was written, an extensive literature on the special theory of
relativity mentioned above has appeared. Let me mention here my address Vber
die geometrischen Grundlagen der Lorentzgruppe, Jahresbericht der deutschen
Mathematiker-Vereinigung, vol. 19 (1910), p. 299, reprinted in Klein's Gesammelte
mathematische Abhandlungen, vol. l, p. 533.

70 Arithmetic: Complex Numbers.

of the three-dimensional space is given by the formula (II) if p and n
have conjugate values, that is, if we write q' = p q p ; or, in expanded
form,

\ ix > + jy > + kz >

I = (d + ia + j b + kc) (ix + jy + kz) (d - ia jb - kc).

In order to prove this, we must show first that the scalar part of the
product on the right vanishes; that is, that q' is indeed a vector. To do
this, we first mutiply p by q according to the rule for quaternion
multiplication, and we find

q' = [ ax by cz + i (dz + bz cy)

+j(dy + ex az) + k (dz + ay bx)] [d ia jb kc] .

After another quaternion multiplication, we actually find the scalar
part of q' to be zero, whereas we find for the components of the vector
part the expressions

x' = (d* + a 2 b 2 c 2 )x + 2(ab cd)y+ 2(ac + bd)z

y' = 2(ba + cd)x + (d 2 + b 2 c 2 a 2 )y h 2(bc ad)z

z' = 2(ca bd)x + 2

(2)

That these formulas actually represent a rotation and expansion becomes
evident if we write the tensor equation for (1), which, by (I), is

x'z + y '2 + 2 '2 _ ( d 2 + a z + b 2 + c 2 ) (x 2 + y 2 + z 2 ) (d* + a 2 + b 2 + c 2 ) ,
or

% '2 JL y'2 + /2 = 7 '4 . (^2 + y 2 + ^2) f

where T ]/d 2 + a 2 -f b 2 + c 2 denotes the tensor of p. Hence, our
transformation is precisely a rotation and expansion (see p. 69), provided
the determinant is positive; otherwise it is such a transformation
combined with a reflection. In any case, the ratio of expansion is M = T 2 .
As remarked above, the determinant must have one of the two values
M 3 = T 6 . If we consider the transformation for all possible values
of the parameters a , b , c , d which correspond to the same tensor value T,
which must obviously be different from zero, we see that the determinant
must always have the value +T Q if it has that value for any single
system of values of a,b 9 c,d m , for the determinant is a continuous
function of a, 6, c, d, and therefore it cannot suddenly change in value
from +T 6 to T 6 without taking on intermediate values. One set
of values for which the determinant is positive is a = b = c =0, d - T 9
since, by (2), the value of the determinant for these values oia,b,c,d,is

d 2 , 0,
0, d 2 ,
0, 0, d 2

Quaternion Multiplication Rotation and Expansion. J\

It follows that the sign is always positive, and hence (1) always re-
presents actually a rotation and expansion. It is easy to write down
a transformation which combines a reflection with a rotation and an ex-
pansion, for we need only combine the preceding transformation with
the reflection x f = x,y' = y,z r = z, which is equivalent to
writing the quaternion equation q' = p q p .

We shall now show that, conversely, every rotation and expansion
may be written in the form (1), or in the equivalent form (2). In the
first place, this formula contains the four arbitrary constants which,
as we saw on p. 69, are' necessary for the general case. That we can
actually obtain any desired value of the expansion-ratio M = T 2 ,
any desired position of the axis of rotation, and any desired angle of
rotation, by a suitable choice of the four arbitrary constants, can be
seen by means of the following formulas. Let f , rj , f denote the direction
cosines of the axis of rotation, and let co denote the angle of rotation.
We have, of course, the well known relation

(3) I s + rf + C 2 = 1 .

I shall now prove that a , b , c , d are given by the equations

d = T cos ~ ;

(4) 2

rn ,. . 0) . 0) rn . .CO

a = 1 sin-- , b 1 Y\ sin - , c = 1 f sin ,

which, by (3), obviously satisfy the condition

d 2 + a 2 + b 2 + c 2 = T 2 .

When these relations have been proved, we can evidently obtain the
correct values of a,b,c,d for any given values of T, , ?j, , co.

To prove the relations (4), let us remark first that if a,b,c,d are
given, the quantities co, ,??, are determined, and in such a way
that (3) is satisfied. For, squaring and adding the equations (4), since T
is the tensor of the quaternion p = d + ia + jb + kc, we have

whence we see that (3) holds. It follows that , i? , C are fully determined
by the relations

which appear directly from (4). These equations express the fact that
the point (a , b , c) lies on the axis of revolution of the transformation.
This fact is easy to verify, for if we put x = a, y = b, z = c in (2),

we find

x' = (d* + a 2 + b 2 + c 2 ) a = T 2 a,

y 9 = (d* + a 2 + Z> 2 + c 2 ) b = T 2 6,
z' == (d 2 + a 2 + b 2 -f c 2 ) c = T 2 - c

72 Arithmetic; Complex Numbers.

that is, the point (a,b, c) remains on the same ray through the origin,
which identifies it as a point on the axis of revolution. It remains
only to prove that the angle co defined by (4) is actually the angle of
rotation. This demonstration requires extended discussion which
I can avoid now by remarking that the transformation (2) for T = 1
reduces precisely to the transformation given by Euler for the revolution
of the axes through the angle co about an axis of revolution whose
direction cosines are , ??, t. This is to be found, for example, in Klein-
Sommerfeld, Theorie des Kreisels, volume 1 x , where explicit mention
of the theory of quaternions is given, or in Baltzer, Theorie und An-
wendung der Determinanten*.

Finally, if we substitute the values given by (4) in the equation (1),
we obtain the very brief and convenient equation in quaternion form
for the revolution through an angle co about an axis whose direction
cosines are ,??,, combined with an expansion of ratio T 2 :

ix' + jy f + kz' = T 2 {cos| + sin ~ (if + jr t + *)}{** + jy + kz}

{CO . CD ,..,. , 7 f.\ 1

cos - - sin -- (t f + 7 r/ + A) j .

This formula expresses in a form that is easy to remember Euler' s
formulas for rotation: the multipliers which precede and follow the
vector ix + jy + kz, are, respectively, the two conjugate quaternions
whose tensor is unity (so-called versor, that is, "rotator", in contra-
distinction to tensor, " stretcher"), and then the whole result is to be
multiplied by a scalar factor which is the expansion-ratio.

We shall proceed now to show that when we specialize these formulas
still further to two-dimensions, they become the well known formulas
for the representation of a rotation and expansion of the xy plane by
means of the multiplication of two complex numbers. (See p. 57.)
For this purpose, let us choose the axis of rotation as the z axis
( = r\ = 0, C = 1). Then the formula (5), for z = z' = 0, may be
written in the form

(5)

ix + jy = T 2 (cos ~ + k sin-^J (ix + jy) (cos -| Asin-^J,

or, upon multiplication with due regard to the rules for products of the
units,

iod + jy = T 2 |cos-(;* + jy) + sm~(jx iy)||cos ~ fcsinyj

.= r 2 |cos 2 ~-(ix-\-jy) + 2sin-^-cos-^-(/# iy) sin 2 -^- (ix + jy)\

= T*{(ix + jy) cosco + (joe iy)smo)}
= T 2 (cosco + ksmco)(ix +jy).

1 Leipzig 1897; 2nd printing, 1914. 2 Fifth edition, Leipzig 1881.

Quaternion Multiplication Rotation and Expansion. 71

If we now multiply both sides by the right-hand factor ( i), we obtain

x' -f- ky' = T 2 (cosco + ksina)) (x + ky),

which is precisely the rule for multiplying two ordinary complex numbers,
and which can be interpreted as a rotation through an angle a) , together
with an expansion in the ratio T 2 , except that we have used the letter k

in place of the usual letter i to denote the imaginary unit ]/ 1.

Let us now return to three-dimensional space, and let us modify
the formula (1) so that it shall represent a pure rotation without an
expansion. To do so, we must replace x', y' f z' by x' T 2 , y' T 2 , z r T 2 ,
that is, we must replace q' by q' T 2 . If we notice that p~^ = \lp ~p/T 2 ,
we may write the formula for a pure rotation in the form

(6) ix' + jy' + kz' = p (ix + jy + kz) p-\

There is no loss of generality if we assume that p is a quaternion whose
tensor is unity, that is,

p = cos ~- + sin (iS + p? + *f), where | 2 + rf + 2 = \ ,
^ 2,

whence we see that (6) results from (5) if T is set equal to unity. The
formula was first stated in this form by Cayley in 1845 1 .

We may express the composition of two rotations in a particularly
simple form, precisely as we did above for four-dimensional space.
Given a second rotation

*x" + jy" + kz" = p' (ix' + jy' + kz') p'~\
where

P' = cos- + sin^ (r + jif + k?)

the direction cosines of the axis of rotation being ', rj 1 ', f ', and the
angle of rotation being a/, we may write

ix" + jy" + kz" =p'-p- (ix + jy + kz)-p~ l - p'~ l

as the equation for the resultant rotation. Hence the direction cosines
of the axis or rotation, I", r\ n ', C", and the angle of rotation, co", for
the resultant rotation, are given by the equation

0" = cos ^ + sin^- (*" + iff' + k?') =p'-p.

We have therefore found a brief and simple expression for the com-
position of two rotations about the origin, whereas the ordinary formulas
for expressing the resultant rotation appear rather complicated. Since
any quaternion may be expressed as the product of a real number

1 On certain results relating to quaternions, Collected Mathematical Papers,
vol. 1 (1889), p. 123. According to Cayley's own statement (vol. 1, p. 586), however,
Hamilton had discovered the same formula independently.

74 Arithmetic: Complex Numbers.

(its tensor) and the versor of a rotation, we have also found a simple
geometric interpretation of quaternion multiplication as the com-
position of the rotations. The fact that quaternion multiplication is
not commutative then corresponds to the well known fact that the
order of two rotations about a point cannot be interchanged, in general,
without changing the result.

If you desire to make a study of the historical development of
the representations and applications of quaternions which we have
discussed, I would recommend to you an extremely valuable report
on dynamics written by Cayley himself: Report on the progress of the
solution of certain special problems of dynamics*.

I shall close with certain general remarks on the value and the
dissemination of quaternions. For such a purpose, one should distinguish
between the general quaternion calculus and the simple rule for
quaternion multiplication. The latter, at least, is certainly of very
great usefulness, as appears sufficiently from the preceding discussion.
The general quaternion calculus, on the other hand, as Hamilton
conceived it, embraced addition, multiplication, and division of
quaternions, carried to an arbitrary number of steps. Thus Hamilton
studied the algebra of quaternions; and, since he investigated also
infinite processes, he may be said to have created a quaternion theory
of functions. Since the commutative law does not hold, such a theory
takes on a totally different aspect from the theory of ordinary complex
variables. It is just to say, however, that these general and far-reaching
ideas of Hamilton have not justified themselves, for there have not
arisen any vital relationships and interdependencies with other branches
of mathematics and its applications. For this reason, the general theory
has aroused little general interest.

It is in mathematics, however, as it is in other human affairs: there
are those whose views are calmly objective; but there are always some
who form regrettable personal prejudices. Thus the theory of quaternions
has enthusiastic supporters and bitter opponents. The supporters, who
are to be found chiefly in England and in America, adopted in 1907
the modern plan by founding an "Association for the Promotion of the
Study of Quaternions* ' . This organization was established as a thoroughly
international institution by the Japanese mathematician Kimura, who
had studied in America. Sir Robert Ball was for some time its president.
They foresaw great, possible developments of mathematics to be secured
through intensive study of quaternions. On the other hand, there are
those who refuse to listen to anything about quaternions, and who go
so far as to refuse to consider the very useful idea of quaternion mul-

1 Report of the British Association for the Advancement of Science, 1862;
reprinted in Cayley's Collected Mathematical Papers, Cambridge, vol. 4 (1891),
pp. 552ff.

Complex Numbers in School Instruction. 75

tiplication. According to the view of such persons, all computation
with quaternions amounts to nothing but computation with the four
components; the units and the multiplication table appear to them to
be superfluous luxuries. Between these two extremes, there are many
who hold that we should always distinguish carefully between scalars
and vectors.

4. Complex Numbers in School Instruction

I shall now leave the theory of quaternions and close this chapter
with some remarks about the role which these concepts play in the cur-
riculum of the schools. No one would ever think of bringing up
quaternions in a secondary school, but the common complex numbers
% + iy always come up for discussion. Perhaps it will be more interesting
if, instead of telling you at length how it is done and how it ought to
be done, I exhibit to you, by means of three books from different periods,
how instruction has developed historically.

I put before you, first, a book by Kastner who had a leading position
in Gottingen in the second half of the eighteenth century. In those
days one still studied, at the university, those elementary mathematical
things which later, in the thirties of the nineteenth century, went over
to the schools. Accordingly, Kastner also gave lectures on elementary
mathematics, which were heard by large numbers of non-mathematical
students. His book, which formed the basis of these lectures, was called
Mathematische Anfangsgrunde*. The portion which interests us here
is the second division of the third part: Anfangsgrunde der Analysis
endlicher Grofien** 1 . The treatment of imaginary quantities begins there
on page 20 in something like the following words: "Whoever demands
the extraction of an even root of a 'denied' quantity (one said 'denied',
then, instead of 'negative'), demands an impossibility, for there is no
'denied' quantity which would be such a power". This is, in fact, quite
correct. But on page 34 one finds: "Such roots are called impossible
or imaginary", and, without much investigation as to justification, one
proceeds quietly to operate with them as with ordinary numbers,
notwithstanding their existence has just been disputed as though, so
to speak, the meaningless became suddenly usable through receiving
a name. You recognize here a reflex of Leibniz's point of view, according
to which, imaginary numbers were really something quite foolish but
they led, nevertheless, in some incomprehensible way, to useful results.

Kastner was, moreover, a stimulating writer; he achieved quite
a place in the literature as a coiner of epigrams. To cite only one of many
examples, he expatiates, in the introduction of this book mentioned

1 Third edition. Gottingen 1794.
* Elements of Mathematics.
** Elemements of Analysis of Finite Quantities.

76 Arithmetic: Complex Numbers.

above, on the origin of the word algebra, which, indeed, as the article
"al" indicates, comes from the Arabic. According to Kastner, an
algebraist is a man who "makes" fractions "whole", who, that is, treats
rational functions and reduces them to a common denominator, etc.
It is said to have referred, originally, to the practice of a surgeon in
mending broken bones. Kastner then cites Don Quixote, who went to
an algebraist to get his broken ribs set. Of course, I shall leave undecided,
whether Cervantes really adopted this form of expression or whether
this is only a lampoon.

The second work which I put before you is more recent, by a whole
series of years, and comes from the Berlin professor M. Ohm: Versuch
eines vollstandig konsequenten Systems der Mathematik* 1 ; a book with
purpose similar to that of Kastner and at one time widely used. But
Ohm is much nearer the modern point of view, in that he speaks clearly
of the principle of the extension of the number system. He says, for
example, that, just like negative numbers, so j/ 1 must be added to
the real numbers as a new thing. But even his book lacks a geometric
interpretation, since it appeared before the epoch-making publication
by Gauss (1831).

Finally, I lay before you, out of the long list of modern school books,
one that is widely used: Bardeys Aufgabensammlung 2 . The principle
of extension comes to the fore here, and, in due course, the geometric
interpretation is explained. This may be taken as the general position
of school instruction today, even if , at isolated places, the development
has remained at the earlier level. The point of view adopted in this
book seems to me to yield the treatment best adapted to the schools.
Withhout tiring the pupil with a systematic development, and without,
of course, going into logically abstract explanations, one should explain
complex numbers as an extension of the familiar number concept, and
should avoid any touch of mystery. Above all, one should accustom
the pupil, at once, to the graphic geometric interpretation in the complex
plane!

With this, we come to the end of the first main part of the course,
which was dedicated to arithmetic. Before going over to similar dis-
cussions of algebra and analysis, I should like to insert a somewhat
extended historical appendix in order to throw new light upon the
general conduct of instruction at present, and upon those features of it
which we would improve.

1 Nine volumes. Berlin 1828. Vol.1: Arithmetik und Algebra, p. 276.

* An Attempt to Construct a Consistent System of Mathematics.

[ 2 See also the Reformausgabe of Bardeys Aufgabensammlung, revised by
W. Lietzmann and P. Ziilke. Oberstufe. Verlag Teubner. Leipzig.] See also
Fine, H., The Number-System in Algebra. Heath. Fine, H., College Algebra.
Ginn.

Development and Structure of Analysis. 77

Concerning the Modern Development and the General
Structure of Mathematics

Let me proceed from the remark that, in the history of the development
of mathematics up to the present time, we can distinguish clearly two
different processes of growth, which now change places, now run side
by side independent of one another, now finally mingle. It is difficult
to put into vivid language the difference which I have in mind, because
none of the current divisions fits the case. You will, however, under-
stand from a concrete example, namely, if I show how one would compile
the elementary chapters of the system of analysis in the sense of each of
these two processes of development.

If we follow the one process, which we will call briefly Plan A,
the following system presents itself, the one which is most widespread
in the schools and in elementary textbooks.

1. At the head stands the formal theory of equations, that is to say,
the operating with rational integral functions and the handling of the
cases in which algebraic equations can be solved by radicals.

2. The systematic pursuit of the idea of power and its inverses yields
logarithms, which prove to be so useful in numerical calculation.

3. Whereas (up to this point) the analytic development is kept quite
separate from geometry, one now borrows from this field, which yields
the definitions of a second kind of transcendental functions, the trigono-
metric functions, the further theory of which is built up as a new separate
subject.

4. Then follows the so called "algebraic analysis' ', which teaches
the development of the simplest functions into infinite series. One considers
the general binomial, the logarithm and its inverse, the exponential func-
tion, together with the trigonometric functions. Similarly, the general
theory of infinite series and of operations with them belongs here. It is
here that the surprising relations between the elementary transcendentals
appear, in. particular the famous Euler formula

e ix = cosx + isin x.

Such relations seem the more remarkable because the functions which
occur in them had been originally defined in entirely separate fields.

5. The consistent continuation, beyond the schools, of this structure,
is the Weierstrass theory of functions of a complex variable, which
begins with the properties of power series.

Let us now set over against this, in condensed form the second
process of development, which I shall call Plan B. Here the controlling
thought is that of analytic geometry, which seeks a fusion of the perception
of number with that of space.

1. We begin with the graphical representation of the simplest functions,
of polynomials, and rational functions of one variable. The point in

78 Modern Development and Structure of Mathematics.

which the curves so obtained meet the axis of abscissas put in evidence
the zeros of the polynomials, and this leads naturally to the theory of the
approximate numerical solution of equations.

2. The geometric picture of the curve supplies naturally the intuitive
source both for the idea of the differential quotient and that of the integral.
One is led to the former by the slope of the curve, to the latter by the
area which is bounded by the curve and the axis of abscissas.

3. In all those cases in which the integration process (or the process
of quadrature, in the proper sense of that word) cannot be carried out
explicitly with rational and algebraic functions, the process itself gives
rise to new functions, which are thus introduced in a thoroughly natural
and uniform manner. Thus, the quadrature of the hyperbola defines the
logarithm

"*dx f

while the quadrature of the circle can easily be reduced to the integral

= arcsm#,

that is, to the inverses of the trigonometric functions. You know that
the same line of thought, pursued farther, leads to new classes of
functions of higher order, in particular to elliptic functions.

4. The development into infinite power series of the functions thus
introduced is obtained by means of a uniform principle, namely Taylor's
theorem.

5. This method carried higher, yields the Cauchy-Riemann theory of
analytic functions of a complex variable, which is built upon the Cauchy-
Riemann differential equations and the Cauchy integral theorem. If we
try to put the result of this survey into definite words, we might say
that Plan A is based upon a more particularistic conception of science
which divides the total field into a series of mutually separated parts and
attempts to develop each part for itself, with a minimum of resources and
with all possible avoidance of borrowing from neighboring fields. Its ideal
is to crystallize out each of the partial fields into a logically closed system.
On the contrary, the supporter of Plan B lays the chief stress upon the
organic combination of the partial fields, and upon the stimulation which
these exert one upon another. He prefers, therefore, the methods which open
for him an understanding of several fields under a uniform point of view.
His ideal is the comprehension of the sum total of mathematical science
as a great connected whole.

One cannot well be in doubt as to which of these two methods has
more life in it, as to which would grip the pupil more, in so far as he is
not endowed with a specific abstract mathematical gift. In order to
bring this home, think only of the example of the functions e x and sin x,

Development of Analysis in the Schools. 79

about which we shall later have much to say along just this line! In
Plan A, which the schools, unfortunately, follow almost exclusively,
both functions come up in thoroughly heterogeneous fashion: e x or,
as the case may be, the logarithm, is introduced as a convenient aid in
numerical calculation, but sin x appears in the geometry of the triangle.
How can one understand, thus, that the two are so simply connected,
and, more, that the two appear again and again in the most widely
differing fields which have not the least to do, either with the technique
of numerical calculation or with geometry, and always of their own
accord, as the natural expression of the laws that govern the subject
under discussion ? How far these possibilities of application go is shown
by the names compound interest law or law of organic growth, which have
been applied to e x , and likewise by the fact that sin x plays a central
role wherever one has to do with vibrations. But in Plan B these
connections make their appearance quite intelligibly, and in accord with
the significance of the functions, which is emphasized from the start. The
functions e x and sin x arise here, indeed, from the same source, the
quadrature of simple curves, and one is soon led from there, as we shall
see later on, to the differential equations of simplest type

de x _ x

-j e , T' J ~O

dx dx*

respectively, which lie naturally at the basis of all those applications.

For a complete understanding of the development of mathematics
we must, however, think of still a third Plan C, which, along side of
and within the processes of development A and B, often plays an
important role. It has to do with a method which one denotes by the
word algorithm, derived from a mutilated form of the name of an Arabian
mathematician. All ordered formal calculation is, at bottom, algorithmic,
in particular, the calculation with letters is an algorithm. We have
repeatedly emphasized what an important part in the development of
the science has been played by the algorithmic process, as a quasi-
independent, onward-driving force, inherent in the formulas, operating
apart from the intention and insight of the mathematician, at the time,
often indeed in opposition to them. In the beginnings of the infinitesmal
calculus, as we shall see later on, the algorithm has often forced new
notions and operations, even before one could justify their admissibility.
Even at higher levels of the development, these algorithmic considera-
tions can be, and actually have been, very fruitful, so that one can justly
call them the groundwork of mathematical development. We must then
completely ignore history, if, as is sometimes done today, we cast these
circumstances contemptuously aside as mere "formal" developments.

Let me now follow more carefully through the history of mathematics
the contrast of these different directions of work, confining myself of course

gO Modern Development and Structure of Mathematics.

to the most important features of the development. The thoroughgoing
difference between A and B, within the whole field of mathematics, will
appear here more clearly than it did above, where our thoughts were
directed only to analysis.

With the ancient Greeks we find a sharp separation between pure and
applied mathematics, which goes back to Plato and Aristotle. Above all,
the well known Euclidean structure of geometry belongs to pure mathe-
matics. In the applied field they developed, especially, numerical calcula-
tion, the so called logistics (Aoyoc = general number, see p. 32). To
be sure, the logistics was not highly regarded, and you know that this
prejudice has, to a considerable extent, maintained itself to this day
mainly, it is true, with only those persons who themselves cannot
calculate numerically. The slight esteem for logistics may have been
due in particular to its having been developed in connection with
trigonometry and the needs of practical surveying, which to some does not
seem sufficiently aristocratic. In spite of this fact, it may have been
raised somewhat in general esteem by its application in astronomy,
which, although related to geodesy, always has been considered one of
the most aristocratic fields. You see, even with these few remarks,
that the Greek cultivation of science, with its sharp separation of the
different fields, each of which was represented with its rigid logical
articulation, belonged entirely in the plan of development A . Nevertheless
the Greeks were not entire strangers to reflections in the sense of Plan B,
and these may have served them for heuristic purposes, and for a first
communication of their discoveries, even if the form A appeared to
them indispensable for the final presentation. This is indicated quite
pointedly in the recently discovered manuscript of Archimedes^, in which
he exhibits his calculations of volume through mechanical considerations,
in a thoroughly modern, pleasing way, which has nothing to do with
the rigid Euclidean system.

Besides the Greeks, in ancient times, the Hindus, especially, played
a mathematical role as creators of our modern system of numerals, and
later the Arabs, as its transmitters. The first beginnings of operating with
letters were made also by the Hindus. These advances belong obviously
to the algorithmic course of development C.

Coming now to modern times, we can, first of all, date the mathematical
renaissance from about 1500, which produced an entire series of great
discoveries. As an example, I mention the formal solution of the cubic
equation (Cardan's formula), which was contained in the "Ars Magna"
of Cardano, published in 1545, in Niirnberg. This was a most significant
work, which holds the germs of the modern algebra, reaching out beyond

1 Cf. Heiberg und Zeuthen, Eine neue Schrift des Archimedes. Leipzig 190?.
Reprint from Bibliotheca Mathematica. Third series, vol. VIII. See also HEATH,
T. L. t The Works of Archimedes. Cambridge University Press.

Brief Survey of the History of Mathematics. 81

the scheme of ancient mathematics. To be sure, this work is not Cardano's
own, for he is said to have taken from other authors not merely his
famous formula but other things as well.

After 1550 trigonometric calculation was in the foreground. The first
great trigonometric tables appeared in response to the needs of astronomy,
in connection with which I will mention only the name of Copernicus.
From about 1600 on, the invention of logarithms continued this develop-
ment. The first logarithmic tables, which a Scotchman Napier (or Nep6r)
compiled, contained, in fact, only the logarithms of trigonometric
functions. Thus we see, during these hundred years,. a path of develop-
ment which corresponds to the Plan A.

We come now, in the seventeenth century, to the modern era proper,
in which the Plan B comes distinctly into the foreground. In 1637
appeared the analytic geometry of Descartes, which supplies the funda-
mental connection between number and space for all that follows. A
reprint 1 makes this work conveniently accessible. Now come, in close
sequence with this, the two great problems of the seventeenth century, the
problem of the tangent, and the problem of quadrature, in other words,
the problems of differentiating and integrating. For the development of
differential and integral calculus, in a proper sense, there was lacking
only the knowledge that these two problems are closely connected, that
one is the inverse of the other. A recognition of this fact was the principal
item in the great advance which was made at the end of the seventeenth
century.

But before this, in the course of the century, the theory of infinite
series, in particular, of power series, made its appearance, and not, in-
deed, as an independent subject, in the sense of the algebraic analysis
of today, but in closest connection with the problem of quadrature. Nicolaus
Mercator (the German name "Kaufmann" latinized; 1620 1687), not
the inventor of the Mercator projection, was a pioneer here. He had
the keen idea of converting the fraction 1/(1 + x) into a series, by dividing
out, and of integrating this series term by term, in order to get the series
development for log (1 + x):

That is the substance of his procedure, although he did not, of course,
use our simple symbols f , dx, . . ., but rather a much more clumsy
form of expression. In the sixties, Isaac Newton (1643 1727) took over
this process, to apply it to the series for the general binomial, which he
had set up. In this process he drew his conclusions by analogy, basing

1 Descartes, R., La Gtomttrie. Nouvtlle Edition. Paris 1886. Translation
by Smith, D. E., and Latham, M. L., 1925. Open Court.

Klein, Elementary Mathematics. 6

arc sm#.

82 Modern Development and Structure of Mathematics.

them on the known simplest cases, without having a rigorous proof
and without knowing the limits within which the series development
was valid. We observe here, again, the operation of the algorithmic

process C. By applying the binomial series to J==L= (1 tf 2 )"" 1 / 2

V i x

C x dx

and using Mercator's process, he gets the series for I f =

Jo V 1 x

By a very skillful inversion of this series, and also of the one for log x ,
he finds the series for sin x and for e x . The conclusion of this chain of
discoveries is due to Brook Taylor (1685 1731) who, in 1715, published
his general principle for developing functions into power series.

As is indicated above, the origin of infinitesimal calculus, at the end
of the seventeenth century, was due to G. W. Leibniz (16461716) and
Newton. The fundamental idea with Newton is the notion of flowing. Both
variables A;, y,aretought of as functions, <p(t), ip(t), of the time t\ and as
the time "flows", they flow also. Newton, accordingly, calls the variable
fluens and designates as fluxion x, y , that which we call differential
coefficient. You see how everything here is based firmly on intuition.

It was much the same with the representation of Leibniz, whose first
publication appeared in 1684. He himself declares that his greatest
discovery was the principle of continuity in all natural phenomena, that
"Natura non facit saltum". He bases his mathematical developments
upon this concept, another example of the Plan B. However, the
influence of the algorithm C is very strong, also, with Leibniz. We get
from him the algorithmically valuable symbols dy/dx and f f (x) dx.

The sum total, however, of this cursory view is that the great discov-
eries of the seventeenth century belong primarily to the plan of develop-
ment B.

In the eighteenth century, this period of discovery continues at first
in the same direction. The most distinguished names to be mentioned
here are L. Euler (1707-1783) and J. L. Lagrange (17361813). Thus
the theory of differential equations, in the most general sense, including
the calculus of variations, were developed, and analytical geometry and
analytical mechanics were extended. Everywhere there was a gratifying
advance, just as in geography, after the discovery of America, the new
countries were first traversed and explored in all directions. But just
as there was, as yet, no thought of exact surveys, just as at first one had
entirely false notions as to the location of these new places (Columbus,
indeed, thought at first, that he had reached Eastern Asia!), just so,
in the newly conquered region of mathematics, that of infinitesimal
calculus, one was, at first, far removed from a reliable logical orientation.
Indeed one even cherished illusions concerning the relation of the calculus
to the older familiar fields, in thatone looked upon infinitesimal calculus
as something mystical that in no way submitted to a logical analysis.

Brief Survey of the History of Mathematics. 8}

Just how untrustworthy the ground was on which the theory stood,
became manifest only when it was attempted to prepare textbooks which
should present the new subject in an intelligible way. Then it became
evident that the method of procedure B was no longer adequate, and it
was Euler who first abandoned it. He had, to be sure, no serious doubts
concerning infinitesimal calculus, but he thought that it caused too
many difficulties and misgivings for the beginner. For this pedagogical
reason he thought it advisable to give a preparatory course, such as
he provided in his text book Introductio in analysin infinitorum (1748),
and which we call today algebraic analysis. To this he relegated, in
particular, the theory of infinite series and other infinite processes, which
he then afterwards used as a foundation in constructing the infinitesimal
calculus.

Lagrange took a much more radical course, nearly fifty years later,
in his Th&orie des fonctions analytiques, in 1797. He could satisfy his
scruples as to the current foundations of infinitesimal calculus only by
discarding it entirely, as a general branch of knowledge, and by consider-
ing it as an aggregate of formal rules applying to certain special classes
of functions. Indeed, he considers exclusively such functions as can be ex-
pressed by means of power series:

fix) = a + a^x + a 2 x 2 + a 3 x* + - - .

He calls these analytic functions, meaning thereby functions which appear
in analysis and with which one can reasonably hope to do something.
The differential quotient of such a function, f (x), is then defined, purely
formally , by means of a second power series, as we shall see later. Diffe-
rential and integral calculus was concerned, then, with the mutual
relations of power series. This restriction to formal consideration ob-
viated, for a time, of course, a number of difficulties.

As you see, the turn which Euler gave, and still more, the entire method
of Lagrange, belongs strictly to the direction A, in that the perceptual genetic
development is replaced by a rigorous closed circle of reasoning. These
two investigators have had a profound influence upon instruction in the
secondary schools, and when the schools today study infinite series, or
solve equations by means of power series according to the so called
method of indeterminate coefficients, but decline to take up differential
and integral calculus proper, they are exhibiting precisely the after effect
of Euler y s "introductio" and of Lagrange' s thought.

The nineteenth century, to which we come now, begins primarily
with a more secure foundation of higher analysis, by means of criteria of
convergence, about which one had hitherto thought but little. The
eighteenth century was the "blissful" period, during which one did
not distinguish between good and bad, convergent and divergent. Even
in Euler 's Introductio, divergent and convergent series appear peaceably

84 Modern Development and Structure of Mathematics.

side by side. But, at the beginning of the new century Gauss (17771855)
and Abel (18021829) made the first rigorous investigations regarding
convergence; and in the twenties Cauchy (17891857) developed, in
lectures and in books, the first rigorous founding of infinitesimal calculus
in the modern sense. He not only gives an exact definition of the differential
quotient, and of the integral, by means of the limit of a finite quotient and
of a finite sum, respectively, as had previously been done, at times; but,
by means of the mean-vahie theorem he erects upon this, for the first
time, a consistent structure for the infinitesimal calculus. We shall come
back to this fully later on. These theories also partake of the nature
of Plan A , since they work over the field in a logical sytematic way,
quite apart from other branches of knowledge. Meanwhile they had no
influence upon the schools, although they were thoroughly adapted to
dispel the old prejudice against differential and integral calculus.

I shall now emphasize only a very little of the further development of
the nineteenth century. In the first place, I shall speak of a few advances
which lie in the direction B: modern geometry, mathematical physics,
along with theory of functions of a complex variable, according to Cauchy
and Riemann. The leaders, in the first working over of these three
great fileds, were the French. This is the place to say a word, also, about
the style of mathematical presentation. In Euclid, one finds everything
according to the scheme "hypothesis, conclusion, proof", to which is
added, sometimes, the "discussion", i.e., the determination of the limits
which the considerations are valid. The belief is widespread that
mathematics always moves thus four steps at a time. But just in the
period of which we are speaking, there arose, especially among the
French, a new art form in mathematical presentation, which might be
called artistically articulated deduction. The works of Monge or, to mention
a more recent book, the Traite d y Analyse, by Picard, read just like a
well written gripping novel. This is the style which fits the method of
thought B, whereas the Euclidean presentation is related, in essence, to
the method A.

Of Germans who achieved distinction in these fields I should mention
Jacobi (18041851), Riemann (18261866), and, coming to a somewhat
later time, Clebsch (18331872), and the Norwegian Lie (1842-1899).
These all belong essentially to the direction B, except that occasionally an
algorithmic touch is noticeable with them.

From the middle of the century on, the method of thought A comes
again to the front with Weierstrass (1815 1 897) . His activity, as teacher
in Berlin, began in 1856. I have already instanced Weierstrass function
theory as an example of A. The more recent investigations concerning the
axioms of geometry belong, likewise, to the type A. One is concerned
here with studies entirely in the Euclidean direction, which approach it,
also, in the manner of presentation.

Brief Survey of the History of Mathematics. 85

With this I bring our brief historical resume to an end. Many points
of view which could only be alluded to here will be brought up later for
more complete discussion. As a summary, we might say that, in the history
of mathematics during the last centuries, both of our chief methods of investiga-
tion were of importance] that each of them, and sometimes the two in suc-
cession, have resulted in important advances of the science. It is certain
that mathematics will be able to advance uniformally in all directions,
only if neither of the two methods of investigation is neglected. May each
mathematician work in the direction which appeals to him most strongly.

Instruction in the secondary schools, however, as I have already
indicated, has long been under the one-sided control of the Plan A.
Any movement toward reform of mathematical teaching must, therefore,
press for more emphasis upon direction B. In this connection I am
thinking, above all, of an impregnation with the genetic method of
teaching, of a stronger emphasis upon space perception, as such, and,
particularly, of giving prominence to the notion of function, under fusion
of space perception and number perception! It is my aim that these
lectures shall serve this tendency, especially since these elementary
mathematical books to which we are in the habit of going for advice,
e g., those of Weber- Wells tein, Tropfke, M. Simon, represent the direc-
tion A almost exclusively. I called your attention, in the introduction,
to this one-sidedness.

And now, gentlemen, enough of these diversions; let us pass to the
next main subdivision of this course.

Part II

Algebra

Let me commence by mentioning a few textbooks of algebra, in order
to introduce you somewhat to a very extensive literature. I suggest,
first, Serret's Cours d'algebre 1 which was much used in Germany, formerly,
and had great merit. Now, however, we have two great widely used
German textbooks: H. Weber's Lehrbuch der Algebra 2 and E. Netto's
Vorlesungen uber Algebra*, each in two volumes; both treat with great
fullness the most difficult parts of algebra and are well adapted for
extensive special study; they seem to me to be too comprehensive for
the average needs of prospective teachers and also too expensive. More
fitting in the latter respect is the handy Vorlesungen uber Algebra* by
G. Bauer, which hardly goes beyond what the teacher should master 5 .
On the practical side, for the numerical solution of equations, this book
is supplemented by the little book Praxis der Gleichungen by C. Runge 6 ,
which I can highly reccomend.

Turning now to the narrower subject, let me remark that I cannot,
in the limits of this course of lectures, give a systematic presentation of
algebra] I can give, rather, only a one sided selection, and it will be most
fitting if I emphasize those things which are, unfortunately, neglected
elsewhere, and which are calculated nevertheless to throw light upon
school instruction. All of my algebraic developments will group them-
selves about one point, namely, about the application to the solution of
equations of graphical and, generally speaking, of geometrically perceptual
methods. This field alone is a very extensive and widely related chapter
of algebra. Even from it, it is obviously possible to select only the most

1 Third edition. Paris 1866 [sixth edition, 1910].

2 Second edition. Braunschweig 1898/99. [New revision by R. Fricke. Vol. I.
1924.]

3 Leipzig 1896/99. See also: Chrystal, Textbook in Algebra (2 volumes).
Macmillan. Bocher, M., Introduction to Higher Algebra. Macmillan.

[ 4 Second edition. Leipzig 1910.]

5 See also: Netto, E., Elementare Algebra, akademische Vorlesungen fur
Studierende der ersten Semester. [Second edition. Leipzig 1913, and H.Weber,
Lehrbuch der Algebra. Small edition in one volume. Second printing. Braunschweig
1921.] See also: Fine, H., College Algebra. Ginn. Hall und Knight, Higher
Algebra. Macmilian.

[ 6 Second edition. Leipzig 1921.] See also: v. Sanden, H., Practical Mathemat-
ical Analysis. Button & Co.

Equations with one Parameter. 7

important and interesting things; in doing this we shall come into
organic relation with the most widely differing fields, so that we shall
be studying mathematics quite in the spirit of our system B. In the first
place, we shall treat equations in real unknowns in order that we may
follow, later, with the consideration of complex quantities.

I. Real Equations with Real Unknowns

1. Equations with one parameter

We begin with a very simple case, which is susceptible of geometric
treatment, namely with a real algebraic equation for the unknown x,
in which a parameter A appears:

/M) = 0.

We shall obtain a geometric representation most simply if we replace A
by a second variable y and think of

f(*,y) = o

as a curve in the xy plane (see Fig. 19). The points of intersection of

this curve with the line y = A, parallel to the x-axis, give the real roots

of the equation / (#, A) = . When we have

drawn the curve approximately, as we can

easily do if / is not too complicated, we can

see at a glance by displacing the parallel

as A varies, how the number of real roots

changes. This plan is especially effective

when / is linear in A, i.e. with equations

of the form

Fig. 19-

<p(x) Ay(#) =

If <p and y are rational, the curve y = y (x)/ip (x) will also be rational, and
is easy to draw. In these cases one can often use this method to ad-
vantage in calculating approximately the roots of equations.
As an example consider the quadratic equation

x 2 + ax A = 0.

The curve y = x* + ax is a. parabola, and one can see at once for what
values of A the equations has two, one, or no real roots according as the
horizontal line cuts the pafabola in two, one, or no points (see Fig. 20).
It seems to me that the presentation of such a simple and obvious con-
struction would be very appropriate in the upper school classes.
As a second example let us take the cubic equation

x* + ax* + bx A = 0,

which gives us the cubical parabola y = x 3 + ax 2 + bx, whose appear-
ance is different according to the values of a , b . In Fig. 21 , it is assumed

Algebra: Real Equations with Real Unknowns.

that x 2 + ax + b = has two real roots. It is easy to see how the
parallels group themselves into those which intersect the curve in one

Fig. 20.

Fig. 21.

point and those which meet it in three ; there can be two limiting positions
which yield double roots.

2. Equations with two parameters

When several parameters, let us say two, appear in an equation,
more skill is required to handle the problem graphically, but the results
are more extensive and interesting. We shall limit ourselves to the
case where the two parameters h , p appear linearly, and we shall write t
for the unknown in the equation. The problem is to determine the real
roots of the equation

(1) 9>W+*'Z0+fvW ='0,
where <p, %, y> are polynomials in t.

If x, y are ordinary rectangular point-coordinates, every straight
line in the x y plane will be given by an equation of the form

(2) y + ux + v = Q.

We may call u, v the coordinates of the straight line. Then ( u) is the
trigonometric tangent of the angle which the line makes with the

tan <p u
Fig. 22.

and ( v) is the y-intercept
(see Fig. 22). Let us think of points
and lines as of equal importance;
and let us give equal attention to
point coordinates and line coordi-
nates. This will be especially impor-
tant later on. Then we may say that
the equation y + MX + v = indicates
the united position of the line (u , v)
that the point lies on the given line,

and of the point (x, y), i. e

and the line goes through the given point.

In order now to interpret the equation (1) geometrically, let us
identify it with (2). This can be done in two essentially different ways
which we shall consider, separably.

Equations with two Parameters.
A. Let us consider the equations

v '

(3b)

If t is variable, the equations (3 a) represent, a witt determined rational
curve of the xy plane, which is called the normal curve of equation (1).
Since every point on it corresponds to a definite value of t, a certain
scale of values of t is defined upon it. By means of (3 a) we can calculate
as many points as we please; and hence we can draw the normal curve,
with its scale, as accurately as we please, say on millimeter paper.
For every definite pair of values of A and p (3b) represents a straight
line of the plane. From what has been said, it follow that (1) shows,
that the point t of the normal curve lies upon this straight line. Thus
we obtain all the real roots of (1) if we find all the real intersections of the
normal curve with this line and read off their parameter values on the
curve scale. The normal curve is determined, once for all, by the form
of equation (1), regardless of the special values which the parameters 4, /a
may have. For every equation with definite A , // there is, then, one
straight line which represents it, in the manner described above, so that,
in general, all the straight lines in the plane
come into play, whereas before (pp. 8788)
only horizontal lines were used.

As an illustration, let us take the quad-
ratic equation

< 2 + J< + A* = 0.

The normal curve here is given by the
equations

y = t 2 , x = t or y = x 2 ,
i.e., the normal curve is the parabola shown
in Fig. 23, with the scale there indicated.

We can at once read off the real roots of our equation as the inter-
sections with the line y + A# + ^ = 0. In particular, the figure
shows that the two roots of the equation t 2 t \ = lie between
f and 2, and between and 1 , respectively. The essential advantage
of this method, over that given on pp. 8788, is that we can now solve
all quadratic equations with one and the same parabola, if we make use
of all the straight lines in the plane. Thus, if we wish to solve, approxi-
mately, a considerable number of equations, one can apply this method
very effectively.

In a similar way one can treat the totality of cubic equations, all of
which can, by a linear transformation, be thrown into the reduced form

t* + it + fi = 0.

Fig. 23.

Algebra: Real Equations with Real Unknowns.

The normal curve here is the cubical parabola

= t or

y =

sketched in Fig. 24. This method also seems to me to be usable in the

schools. The pupils would certainly derive pleasure from drawing such

curves.

B. The second method of interpreting (1) is got from the first by

applying the principle of duality, i. e., by interchanging point and line

coordinates. To that end, let us write
the terms of (2) in reverse order:

v + %u + y =

and identify it, in this form, with (1)
by setting

/ W

< /w v /

(4a)
(4b)

Y I A; ii

x A , y ft, .

If t is variable, the equation (4 a)
represents a family of straight lines
which will envelope a definite curve,
Fig. 24. the normal curve of (1), in the new

interpretation. It is a rational class

curve, since it is represented, in line coordinates, by rational functions of
a parameter. Every tangent, and hence the corresponding point of tan-
gency, is determined by a definite value of t, so that one gets again a
scale on the normal curve. By drawing a sufficient number of tangents
according to (4 a), we may draw both curve and scale with any desired
degree of exactness. Each parameter-pair A , /u, determines, by virtue
of (4b), a point in the xy plane, through which, by virtue of (1), the
tangent t of the normal curve (4 a) must pass. We obtain, therefore, all
the real roots of (\) by reading off the parameter-values t belonging to all
the tangents to the normal curve which go through the point x = A , y p .
As before, the normal curve is completely determined by the form of
equation (1). Every equation of this form will be represented, for given
values of the parameters i , // , by a certain point in the plane, or, if
we wish, by its position with respect to the curve.

Let us illustrate by means of the same examples as before. Corres-
ponding to the quadratic equation

P + It + /LI =
the normal curve will be the envelope of the straight lines

This envelope, again, is a parabola with its vertex at the origin. The
graph, drawn on fine cross section paper exhibits immediately the real

Equations with two Parameters.

roots of t 2 + It + ft = as parameters t of the tangents drawn to the
parabola from the point A,^ (see Fig. 25).
For the cubic equation

, / 3 + A* + ^-0

the normal curve

v = t*, u = t

will be a curve of the third class with a cusp at the origin, shown in
Fig. 26.

We can present this method somewhat differently. If we examine
the so-called trinomial equation

t + lp + p^Q,

we may represent the system of tangents to the normal curve by means
of the parameter equation

f(t) =^ + ^ + y =,:0.

Fig. 25.

The equation of the normal curve in point coordinates may be found,
as usual, by eliminating t between the last equation and the equation
obtained by differentiation with respect to t\

/' (t) = mt m ~ l + nxt n - l =

for the normal curve, as the envelope of the system of straight lines,
is the locus of the intersection of each of these lines with the neighboring
line (for t and t + dt). If, instead of eliminating /, we express x and y
as functions of t from these two equations, we find

(5 a)

y -

which are the point equations of the normal curve:

As normal curves for the quadratic and the cubic equations which
were selected above as examples, one finds in this way, respectively,

These are the curves which are sketched in Figs. 25 and 26.

Algebra: Real Equations with Real Unknowns.

Fig. 27.

Let me emphasize the fact that this method is put to practical use
by C. Runge, in his lectures and exercises, and that it has proved itself
especially appropriate for the actual solution of equations. We might
profitably use one or the other of these graphical methods in school
instruction.

If we now compare with each other the two methods which we have
developed, we find that, for at least one definite and very important
purpose, the second offers a distinct advantage, namely, when one seeks
a visible representation of all the equations of a definite type which have
a given number of real roots. Such totalities are represented, according
to the first method, by systems of straight lines ; according to the second,
however, by fields of points. But because of the peculiar nature of our

geometric perception, or of our habit, the
latter are essentially easier to grasp than are
the former.

I shall show at once, by means of the
example of the quadratic equation, what can be
done in this direction (see Fig. 27). From all
points outside of the parabola two tangents
can be drawn to the curve; from points
within, none. Hence these two regions represent
the manifolds of all equations with two roots and with no roots, respectively.
For all points of the parabola itself there is only a single tangent, which
can be counted twice. The normal curve itself is, then, in the general case,

the locus of those points whose coordinates I , ft
yield equations with two equal roots, so that
we may call it the discriminant curve.

In the case of the cubic equation, we see
that from a point inside the angle of the
1 normal curve one can draw three tangents
to the curve. This is obvious for points on
the median line, because of symmetry; and
the number cannot change when the point
varies, provided it does not cross the curve.
If the point (x, y) moves to the curve, two
of the tangents coincide; if it moves into

the region outside the curve, both of these tangents become imaginary
and there remains but one real tangent. Accordingly, the region inside
the angle of the normal curve represents the totality of cubic equation with
three different real roots ; the region outside, equations with only one real
root', while to the points on the curve itself correspond the equations with
one simple and one double real root. Finally, a triple tangent goes
through the cusp, corresponding to the single equation / 3 = 0, with a
single triple root. Figure 28 makes this obvious at a glance.

Fig. 28.

Equations with two Parameters.

The pictures become much more interesting and show more, if, as
is customary in algebra, we impose definite restrictions upon the roots,
in particular, if we inquire about all the real roots lying within a given
interval t l <it^t 2 . As you know, the general answer to this question
is furnished by Sturm's theorem. We can, however, easily complete our
drawing so that it will give a satisfying graphical solution of this general
question also. For this purpose we simply add to the normal curve the
tangents to it determined by the parameter values t lf t 2 and consider the
division of the plane into fields which these tangents bring about.

To carry through these considerations for the quadratic equation,
we must determine the number of tangents which touch the parabolic arc
between t and t 2 . Through every point of the triangle (see Fig. 29)
bounded by the parabolic arc and these two
tangents there are obviously two tangents.
If the point crosses either of the tangents
t lf t 2 , one of the tangents through it will
touch the parabola beyond the arc (t, t 2 ),
and so will be lost for our purpose. Tangents
from points which lie within the two crescent
shaped areas bounded by the parabola and
the tangents t lt t 2 touch the parabola outside
the arc (t t 2 ) ; and from points within the parabola there are no real
tangents at all. The two parabolic arcs t ^ t and t^>t% are thus of
no significance in effecting the desired subdivision of the plane. There
remain, then, only those lines in the figure
which are drawn full; these, together with
the numbers assigned to them, give at a
glance exact information as to the manifolds
of quadratic equations which have 2, 1, or
real roots between t and t 2 .

We may proceed similarly with the cubic
equation (see Fig. 30). Let us take, say, t > 0,
/ 2 < . Again we draw the tangents with these
parameter values and examine the subdivi-
sions of the plane brought about by them
and the arc of the normal curve which lies

between ^ and t 2 . Through every point in the four-cornered region at
the cusp there will be three real tangents which touch between t and t 2 .
If point crosses either of the tangents ^ , t 2 , there is a loss of one tangent
of this character. When it crosses the normal curve two are lost. From
these considerations we obtain the picture, shown in Fig. 30, of the
regions of the plane which correspond to equations with three, two, one,
or no roots lying between t and t 2 . In order to see the great usefulness
of the graphical method, one need only make a single attempt to picture

Fig. 30.

Q4 Algebra: Real Equations with Real Unknowns.

abstractly this classification of cubic equations, without making any
appeal whatever to space perception ; it will require a disproportionately
great amount of time. And the proof, which here becomes evident by
a glance at the picture, will not be at all easy.

Now as to the relation of this geometric method to the well known
algebraic criteria of Sturm, Cartesius, and Budan-Fourier I remark,
merely, that the geometric method includes them all, for equations of
the types which we have considered. You will find these relations
carried out more fully in my article 1 "Geometrisches zur Abzdhlung der
Wurzeln algebraischer Gleichungen" and in W. Dycks "Katalog mathe-
matischer Modelle" 2 . I am glad to take this occasion to refer you to
this catalog. It was published on the occasion of the exposition, in
Munich, in 1893, by the German Mathematical Society, and remains
today the best means of orientation in the field of mathematical models.

3. Equations with three parameters A, fi, v

Finally, I shall also show you that one can apply analogous considera-
tions to equations with three parameters. We shall need to use space
of three dimensions instead of the plane. It will suffice if I consider the
special equation of four terms

The method of procedure can be applied immediately to equations of
other forms.

In addition to this equation, we shall use the condition, from space
geometry, that a point (x , y , z) and a plane with the plane coordinates
(u,v,w) shall be "in united position", i.e., that the plane (u,v,w)
shall contain the point (x,y t z). This condition is

(2) z + u% + vy + w =
or

(3) w + xu + yv + z = 0.

We now identify this equation, written in the one form or the other,
with (1) and we obtain, exactly as before, two mutually dual inter-
pretations.

Let us then set
(2 a) z = P, oc = t m , y = t n .

These equations determine a certain space curve, the normal curve of
the four -term equation (1), together with a scale of the values t. Then we

[* Reprinted in Klein, F., Gesammelte Mathematische Abhandlungen, vol. II,
pp. 198-208.]

2 A catalogue of mathematical and mathematical-physical models, apparatus,
and instruments (Munich, 1892), also a supplement to this (Munich, 1893).

Equations with three Parameters A, ,a, v.

consider the plane which is determined by the coefficients A, fi, v ,

of (1):

(2b) 71 = A, v = /LI, w v.

Then equation (1) says that the real roots of the proposed equation are
identical with the parameter values t of the real intersections of the normal
curve (2 a) with the plane (2b).

If we choose the method dual to the preceding, we must put

(3a) w = t p , u = t m , v = f.

These equations represent, for
variable t, a simple infinity of
planes, which we can look upon
as the osculating planes of a
definite space curve associated,
as before, with a scale of para-
meter values t. This will be a
normal f< class curve" , being ex-
pressed in plane coordinates, in
distinction from the previous
normal "order curve'', which
was given in point coordinates.
If we now consider, in conjunc-
tion with the first curve, also
the point

(3b) x = l, y = t*, z = v,

it follows that the real roots of (1)
are identical with the parameter
values t of those osculating planes
of the normal class curve (3 a)
which pass through the point (3 b).

Let us next illustrate these two interpretations by concrete examples.
We have, in our collection, models for both of them, which I shall now
put before you.

The first method was used by R. Mehmke, in Stuttgart, in the con-
struction of an apparatus for the numerical solution of equations. His
model is a brass frame (see Fig. 31) in which you will notice three
vertical rods carrying scales, and into which one can fit curved templates,
or stencils, of the normal curves of equations of degree three, four, or
five, (after these have been reduced to four terms). Note, however, that
while our exposition presupposed the ordinary rectangular coordinate
system, Mehmke has so determined his coordinate system that the appro-
priate plane coordinates, i.e., the coefficients u, v, w of the equation of
the plane (2), are precisely the intercepts which this plane makes on the

96 Algebra: Real Equations with Real Unknowns.

scales of the three vertical rods and which one can read off there. In
order, now, to make possible the fixation of a definite plane u = A,
v = /LI , w = v, a peep-hole is provided on the w rod, which one sets
at the reading v of that scale, while one joins by a stretched string the
readings, of the scales on the u and v rods, respectively. The rays joining
the peep-hole with this string make our plane, and by looking through
the peep-hole one can observe directly the intersections of this plane with
the normal curve as the apparent intersections of the string with the template.
Their parameter values, the desired roots of the equation, are read at the
same time on the scale of the normal curve, which is affixed to the template.
The practical usableness of this apparatus depends, of course, upon
the carefulness of its mechanical construction, but the limited power
of accommodation of the human eye would, at best, make it very
doubtful.

For the second method, a model was prepared by Hartenstein in con-
nection with his state examination. It has to do with the so-called
reduced form of the equation of degree four, that is,

t* + U^ + iJit + v=Q,

to which every biquadratic equation can be reduced. I shall present
this method in a form somewhat different from the one I used for the
two-parameter equation (p. 91). In the present case we have to consider
a simple infinity of planes whose plane coordinates are given in (3 a)
and whose point equations would be written as follows:

(4) /(*) =P + xt* + yt + * = 0.

The envelope of these planes is the system of the straight lines in which
each plane / (t) = meets the neighboring plane / (t + dt) = 0, i.e.,
the developable surface whose equation is obtained by eliminating t between
f (t) = and /' (t) = . But in order to obtain the normal curve we must
seek the osculating configuration of the system of planes, i.e., the locus
of the points of intersection of three successive planes. This locus is, as
you know, the cuspidal edge of that developable surface and its coordinates
are found, as functions of t , from the three equations f (t} = , /' (t) = ,
f" (t) = 0. In our case these three equations are:

t* + xt 2 + yt + z --= Q
4t* + x-2t + y =
12* a + *-2 =0,

and one finds from them:

(5) x = -6t*, y = 8t*, * = -3.

These expressions represent the point equation of the normal class curve
of (4) whose plane equation, by (3 a), may be written in the form

(6) w = t*, u = t*, v = t.

Equations with Three Parameters A, fi, v.

Both forms are of degree four in t. Hence the normal curve is both of
order four and of class four.

In order to study it more in detail, let us consider a few simple
surfaces which pass through it. In the first place, the expressions (5)
satisfy identically in t the equation

Hence our normal curve lies upon a parabolic cylinder of order two
whose generators are parallel to the y-axis. Likewise, we have the relation

~8~ ~*~ "27 ~~ '

so that this cubic cylinder, whose generators are parallel to the z-axis,
also goes through our normal curve. Moreover, the normal curve is the
finite intersection of these two cy-
linders. With these facts in mind,
one can form an approximate
picture of the course of the nor-
mal curve. Is is a skew curve,
symmetric to the x z plane, having
a cusp at the origin (see Fig. 32).
Again the quadric surface

x-z _ 3y 2 _

6 64

goes through our normal curve;
for, by (5), this equation is also Fig. 32.

satisfied identically in /. From

it, and the equation of the cubic cylinder, we find another linear
combination which represents an especially important surface of the
third degree passing through the normal curve:

^? Z_ 2 _ 5 3 _ _ o
6 16 216 ~~

Let us now consider the developable surface whose cuspidal edge is
the normal curve, and which we can define as the totality of the tangents
to the normal curve. The tangent at the point t to any space curve

is given by the equations

* = v(*) + Q<P' W > y = v W + e W , * = * W + ez'W

in which Q is a parameter. For the direction cosines of the tangents
to the curve are to each other as the derivatives of the coordinates
with respect to / . lit is thought of as variable, we have in these equations,
with two parameters t, Q , the representation of the developable surface.

Klein, Elementary Mathematics. 7

Qg Algebra: Real Equations with Real Unknowns.

All this follows from well known theorems of space geometry. For our
curve (5) we get, in particular, the following equations for the developable
surface. If we call the coordinates of its points (X , Y , Z) to distinguish
them from the coordinates of the curve, the equations of the develop-
able are

(7)

Now this surface is the basis of the Hartenstein model, its straight lines
being represented by stretched threads (see Fig. 33)-

The parameter representation offers the best starting point for the
discussion and the actual construction of the surface. Indeed, it is only
from force of habit that we inquire about the equation of the surface
itself. We can obtain it by eliminating Q and t from (7). I shall give
you the simplest procedure for this without giving the details of the
inner meaning of the several steps. From (7) we form the combination

X 7 V 2 Jf 3
A lf __ J*_ ___ ^L

6 16 216

both of which vanish on the curve itself (for Q = 0). If we equate
these to zero, we obtain two of the surfaces mentioned above which
pass through the curve. Eliminating the product Qt from these equations,
we find the equation of the developable surface

( 7J _X*\* ^( x ' z y2 x3 Y*-n

r + i2J ~~ 27 r 6~ ~ 16 ~ iiej " -

The surface is thus of order six; but it is composed of the plane at in-
finity and a surface of order five.

As to the meaning of this formula, I make the following remark for
those who are acquainted with the subject. The expressions in the two
parentheses are the invariants of the biquadratic equation

t* + Xt*+ Yt + Z = 0,

with which we started. These play an important role in the theory of
elliptic functions and they are designated there, in general, by g 2 an( i 3-
The left side of the equation of our surface, A = gij 27 gL is, as you
know, the discriminant of the biquadratic equation, which indicates, by
its vanishing, the presence of a repeated root. Our developable surface
is therefore the discriminant surface of the biquadratic equation, i.e., the
totality of the points for which it has a double root.

After these theoretical explanations, the construction of a thread
model for our surface offers no essential difficulty. By means of the
parameter equations (7) we may determine, say, the points in which

Equations with Three Parameters

those tangents which we wish to represent intersect certain fixed planes.
We then stretch threads between these planes, which are made out of
wood or cardboard. But it requires long trial and great skill to make
the model really beautiful and usable, and to bring out the entire inter-
esting course of the surface and of its cuspidal edge, as in the model
before us. The sketch on page 99 (see Fig. 33) shows the surface with
its straight lines; AOB is the cuspidal edge [see the figure p. 97 1 ].

You notice on the model a double curve (COD)
along which two sheets of the surface intersect. This
curve is simply the following parabola of the
X Z plane :

Only one half (CO) of this parabola, namely that
for X < 0, appears, however, as the intersection
of real sheets, while the other half lies isolated in
space. This phenomenon is by no means sur-
prising to those who are accustomed to illustrate
the theory of algebraic surfaces by concrete geo-
metric representations. It is a common thing,
there, for real branches of double curves to appear
both as intersections of real sheets and also in
part isolated. In the latter
case we regard them as real
intersections of imaginary sheets
of the surface. The correspond-
ing phenomenon in the plane
is more generally known. In
that case, in addition to the
ordinary double points of al-
gebraic curves, which appear
as intersections of real bran-
ches of the curve, there are also the apparently isolated double points,
which may be regarded as the intersections of imaginary branches.
Let us now make clear in detail, what this surface with its cuspidal
edge, the normal curve, can do for us. We think of the normal curve
with its associated scale, or, better, we affix to each tangent its para-
meter value /, which also belongs to the point of tangency. //, now,
someone gives us a biquadratic equation with definite coefficients (x,y,z),
we need only to pass through the corresponding point (x,y,z)the osculating
plane to the normal curve, or, what would be the same thing, the tangent

1 The Hartenstein string model was put upon the market by the firm of M. Schil-
ling in Leipzig. A dissertation by R. Hartenstein entitled: Die Discriminanten-
fldche der Gleichung vierten Grades goes with the model Leipzig, Schilling, 1909.

Fig. 33.

1QO Algebra: Real Equations with Real Unknowns.

plane to the discriminant surface, to obtain the real roots as the parameter
values of the points of contact with the curve, or the parameter values of
the corresponding tangents, as the case may be. Since the osculating plane
cuts the curve where it touches it, every point of contact of an osculating
plane with the curve is projected from the point (x, y , z) as an apparent
point of inflexion on the curve, and conversely. Consequently, the real
roots of the biquadratic equation are, finally, the parameter values t of
the apparent inflexion points of the normal curve, viewed from the point
(x, y , z) in space.

Now it is, of course, quite difficult for the unpractised eye to deter-
mine with certainty from the model either the planes of osculation or
the apparent inflexions of the curve. But the model exhibits with
immediate clearness the next important thing, the classification of all
biquadratic equations according to the number of their real roots. Let us
see, by ^n abstract examination of equations, just what cases one might
expect . If <x , /? , y , d are the four roots of the real biquadratic equation (4) ,
then & + /? + y + ^=0, because of the vanishing of the coefficient
of / 3 . So far as the reality of the roots is concerned, the following
principal cases are possible:
I. Four real roots.
II. Two real, and two conjugate complex roots.

III. No real, and two pairs of conjugate complex roots.

If, now, two equations of the type I are proposed, with roots a, , /?, y , <5
and <*', /?', y', 6', respectively, then one certainly could transform a, /J,
y, d continuously into <x', /?', y', <5', respectively, through systems of
values whose sum is always zero, At the same time, the one equation
would transform continuously into the other, through equations always
of the same type, i.e., all equations of type I make up a connected
continuum, and the same is true for the other two types. Our model
must therefore exhibit space partitioned into three connected parts such that
the points in each part correspond to equations of one type.

Let us now consider the transition cases between these three sorts.
Type I goes over into II through equations which have two different real
roots and one double (i. e. two coincident) real root, which we shall indicate
symbolically by 2 + (2) ; similarly we have between II and III the
transition case of one real double root and two complex roots, which may
be indicated by (2). To both of these sorts there must correspond, in our
model, regions of the discriminant surface, which, indeed, pictures all
equations with coincident roots. Considerations similar to those above
would show that to each type there must correspond a connected region of
this surface. Now, again, these two groups, 2 + (2) and (2), go over
into each other by means of cases with two real double roots, symbolically :
(2) + (2); the points for which two pairs of roots move thus into co-
incidence must belong simultaneously to two sheets of the discriminant

The Fundamental Theorem of Algebra. -101

surface, that is, to the non isolated branch of the double curve. Accordingly,
the discriminant surface falls into two parts, separated by a branch of the
double curve] one of these parts, 2 + (2), separates the space regions I
and II, the other, (2), the space regions II and III. In order to see, now,
how the normal curve lies, we notice that, because of its property as
a cuspidal edge, three tangent planes must merge into one (the osculating
plane) at each point on it, so that we have the case of a triple and a
simple real root: 1 + (3) . This can happen only when one of the simple
roots becomes equal to the double root. Consequently, the cuspidal edge
must lie entirely on the first part, 2 + (2), of the surface. In the cusp of
the cuspidal edge (x = y = z = 0) we have a quadruple real root, which
can arise from the case (2) + (2) through the coincidence of the two
double roots. In fact, the cusp, , of the cuspidal edge lies also on the
double curve. Finally, as to the isolated branch of the double curve, it lies
entirely in the space region III and is characterized by the fact that
on it the two pairs of conjugate complex roots merge into one complex
double root. Both double roots are, of course, conjugate to each other.
You can recognize on our model all of the possible cases enumerated
above. In the sketch (Fig. 33> P- 99), the interior of the surface to
the right of the double curve is region I, to the left, region III; the
exterior is region II. You will be able easily to become fully oriented
by means of the following tabulation, which exhibits the number and
the multiplicity of the real roots which correspond to the points of the
several space, surface, and curvilinear regions. In this scheme, the digits
not in parentheses denote the number of simple real roots, the others,
as before, denote the multiplicity of repeated roots:

I. II. III.

Region :
Discrim. surface:
Normal curve:
Double curve
Cusp :

4 2

2+ (2)
1 -H3)

(2)

(2) + (2)

(2imag. double roots).

(4)

II. Equations in the field of complex quantities

We shall now remove the restriction to real quantities and shall
operate in the field of complex quantities. Of course, we shall endeavor
again only to emphasize those things which are susceptible of geo-
metric representation to an extent greater than one finds elsewhere.
Let us begin at once with the most important theorem of algebra.

A. The fundamental theorem of algebra
This is, as you know, the theorem that every algebraic equation of
degree n in the field of complex numbers has, in general, n roots, or, more

102 Algebra: Equations in the Field of Complex Quantities.

accurately, that every polynomial f (z) , of degree n, can be separated into n
linear factors.

All proofs of this theorem make fundamental use of the geometric
interpretation of the complex quantity x + iy in the x y plane. I shall
give you the train of thought of Gauss' first proof (1799), which can be
presented quite graphically. To be sure, the original exposition of Gauss
was somewhat different from mine.

Given the polynomial

./(*) = z n + a lZ n - l + ... + a n ,
we may write

f(x + iy) = u (x, y) + i v (x, y) ,

where u , v are real polynomials in the two real variables x . y . The
leading thought of Gauss' proof lies now in considering the two curves

u (x , x} = and v (x , y) =

in the x y plane, and in showing that they must have one point, at least,
in common. For this point one would then have / (x + iy) = 0, that
is, the existence of a first "root" of the equation f = would be proved.
For this purpose, it turns out to be sufficient, to investigate the be-
haviour of both curves at infinity, i.e., at a distance from the origin
which is arbitrarily great.

If r , the absolute value of z, is very large, we may neglect the lower
powers of z in / (z) , in comparison with z n . If we introduce polar co-
ordinates r, <p into the x y plane, i. e., if we set

z = r (cos (jp + i sin q>) ,
we have, by De Moivre's formula

z n = r n (cosnq) + isinwep).

This expression is approached asymptotically by / (z) , as z increases
in absolute value. It follows at once that u and v approach, respectively,
asymptotically the functions

r n cosn(p, r n smn<p.

Consequently the ultimate course of the curves u = 0, v = 0, at in-
finity, respectively, will be given approximately by the equations

cos n <p = , sinn<p = 0.

Now the curve sin n q> = consists of the n straight lines which
go through the origin and make with the #-axis the angles 0, nfn,
2 7i In , . . . , (n 1) n/n , whereas cos n<p = Q consists of the n rays through
the origin which bisect these angles (Fig. 34 is drawn for n = 3). In
the central part of the figure, the true curves u = 0, v = can, of
course, be essentially different from these straight lines; but they must
approach the straight lines asymptotically as the lines recede from the

The Fundamental Theorem of Algebra.

103

origin. We can indicate their course schematically by retaining the straight
lines outside of a large circle and replacing them by anything we please,
inside the circle (see Fig. 35). But no matter what the behavior of the

Fig. 34.

curves may be inside the circle, it
is certain that, if one makes the
circle about the origin sufficiently
large, the branches u , v , outside
the circle, must alternate, from
which it is graphically clear that
these branches must cross one another
inside the circle. In fact, we can
give a rigorous 1 proof of this
assertion, and this is the sub-
stance of Gauss' proof if we use
the continuity properties of the
curves. The preceding argument,
however, gives the essentials of
the train of thought. If one such
root has been found, we can divide
out a linear factor, and we can then

Fig. 36.

1 It should be said here that Gauss does not dispense entirely with geometric
considerations. The arithmetization of the proof which he contemplated in his
dissertation was first given by A. Ostrowski (Gottinger Nachrichten, 1920, or
vol. VIII of the materials for a scientific biography of Gauss, 1920). It is of
historical interest that the first proof of the fundamental theorem was by D'Alem-
bert. To be sure, there was an error in his proof, to which Gauss called attention.
D'Alembert, namely, failed to distinguish between the upper limit of a function
and its maximum, and he made use of the assumption, which in general is false,
that a function of a complex variable actually attains its upper limit when this
limit exists. N

1Q4 Algebra: Equations in the Field of Complex Quantities.

repeat the reasoning for the other polynomial factor of degree (n 1).
Continuing in this way, we may finally break up f (z) into n linear factors,
i. e., we may prove the existence of n zeros.

This method of reasoning will be much clearer if you carry through
the construction for special cases. A simple example would be

/ ( z ) = z * - \ = .

In this case we obviously have

u = ? 3 cos399 1 , v = r 3 sin3<p,

so that v = consists simply of three straight lines, while u = has
three hyperbola-like branches. Figure 36 shows the three intersections
of the two curves, which give the three roots of our equation. I re-
commend strongly that you work through other and more complicated
examples.

These brief remarks about the fundamental theorem will suffice
here, since I am not giving a course of lectures on algebra. Let me
close by pointing out that the significance of the admission of complex
numbers into algebra lies in the fact that it permits a general statement
of the fundamental theorem. With the restriction to real quantities
one can only say that the equation of degree n has n roots, or fewer,
or perhaps none at all.

B. Equations with a complex parameter
The rest of the time which I have set aside for algebra I shall devote
to the discussion, by graphical methods, of all the roots (including the
complex ones] of complex equations, as was done earlier for the real roots
of real equations. We shall limit ourselves, however, to equations with
one complex parameter and we shall assume, furthermore, that this
occurs only linearly. The study of a simple con formal representation will
then give us all that is required.

Let z = x + iy be the unknown, and w = u + iv the parameter.
Then the type of the equation to be considered has the form

(1) <p(z) w-ip(z) =

where (p , y , are polynomials in z . Let n be the highest power of z that
occurs. According to the fundamental theorem, this equation has for
each definite value of w exactly n roots z which, in general, are different.
Conversely, however, it follows from (1) that

i.e., w is a single-valued rational function of z, and it is said to be of
degree n. If we should use, as geometric equivalent of equation (1),

Equations with a Complex Parameter.

simply the conformal representation which this function sets up between
the 2-plane and the z^-plane, the many-valuedness of z as function of w
would be visually disturbing. We may help ourselves here, as is always
the case in function theory, by thinking of the w-plane as consisting
of n sheets, one over another, which are united in an appropriate manner,
by means of branch cuts, into an n leaved Riemann surface. Such surfaces
are familiar to you all from the theory of algebraic functions. Then our
junction establishes, between the points of the n-leaved Riemanris surface
in the w-plane and the points of the simple z-plane, a one-to-one relation
which is, in general, conformal.

Before we begin a detailed study of this representation, it will be
helpful if we set up certain conventions which will do away with the
exceptional role played by infinite values of w and z, a role not justified
by the nature of the case, and which will enable us to state theorems
in general form. Inasmuch as these conventions are not so widely
employed as they should be, you will permit me to say a word or two
more about them than I otherwise should. We cannot be satisfied here
when one speaks merely symbolically of an infinitely distant point of the
complex plane, since such a conception gives no adequate concrete
image, so that one must have recourse to special considerations or stipula-
tions, in order to find out what corresponds, for an infinitely distant
point, to a definite property of a finite point. But we can secure all that
is desired, if we replace the Gaussian
plane, as picture of the complex num-
bers, once for all, by the Riemannian
sphere. For this purpose, we think
simply of a sphere of diameter one,
tangent to the % y plane, its south
pole S being at the origin, and
from its north pole N we project
the plane stereographically upon
the sphere (see Fig. 37). To every
point Q = (x, y) of the plane there
corresponds uniquely the second Fig. 37.

intersection P of the ray NQ

with the sphere; and, conversely, to every point P of the sphere,
with the exception of N itself, there corresponds a unique point Q
with definite coordinate (x, y). Hence we can consider P as representing
the number x + iy. Now if P approaches the north pole N, in any
manner, Q moves to infinity ; conversely, if Q recedes to infinity in any
manner, the corresponding point P approaches the single definite
point N. It seems natural, then, to look upon this point N, which does
not correspond to any finite complex number, as the unique representative
of all infinitely large x + iy, i.e., as the concrete picture of the infinitely

406 Algebra: Equations in the Field of Complex Quantities.

distant point of the plane, which is otherwise introduced only symbolically,
and to affix to it outright the mark <x>. In this way we bring about, in
the geometric picture, complete equality between all finite points and the
infinitely distant point.

In order to return now to the geometric interpretation of the
algebraic relation (1), we shall replace the w plane also by a w-sphere.
Then our function will be represented by a mapping of the z-sphere
upon the w-sphere, and, just as in the case of the mapping of the
two planes, this is also conformal, since the stereographic mapping
of the plane upon the sphere is, according to a well known theorem,
conformal. To a single position on the w- sphere, there will then
correspond, in general, n different positions on the 2-sphere. In order
to get a one-to-one relation we imagine, again, n sheets on the
z0-sphere, lying one above another, and united, in appropriate manner,
by means of branch cuts, so as to form an n-leaved Riemann surface
over the w-sphere. This picture presents no greater difficulty that that
of the Riemann surface over the plane. Thus, finally, the algebraic
equation (1) is interpreted as a one-to-one relation, conformal in general,
between the Riemann surface over the w-sphere and the simple surface
of the z-sphere. This interpretation obviously takes into account, also,
infinite values of z and w which may correspond to each other or to
finite values.

In order to make the greatest possible use this geometric device,
we must take a corresponding step in algebra, one which shall do away
with the exceptional role which infinity plays in the formulas, and this
step is the introduction of homogeneous coordinates. We set, namely,

and consider z^ , z 2 as two independent complex variables, both of which
remain finite, and which cannot both vanish simultaneously. Each definite
value of z will then be given by infinitely many systems of values
(cz lt cz^, where c is an arbitrary constant factor. We shall look upon
all such systems of values (cz lt cz 2 ) which differ only by such a factor,
as the same "position" in the field of the two homogeneous variables.
Conversely, for every such position there will be a definite value of z,
with one exception : to the position (^ arbitrary, z 2 = 0) there will
correspond no finite z; but if one approaches it from other positions,
the corresponding z becomes infinite. This one position is thus to be
looked upon as the arithmetic equivalent of the one infinitely distant point
of the z-plane or, as the case may be, of the z-sphere, and as carrying the
mark z = oo.

In the same way, of course, we put also w = w l /w 2 . We shall now set
up the "homogeneous" equation between the "homogeneous" variables z l9

Equations with a Complex Parameter. 107

z 2 and w lf w 2 , which corresponds to equation (2). Multiplying by z% in
order to clear of fractions, we may write the equation in the form

In this equation, q> (z l , 2 2 ) and y (^ , 2 2 ) are rational integral functions
of z and, z z , since 9? (2) and^ (z) contain at most the nth power of z = z l /z 2 -
Moreover they are homogeneous polynomials (forms) of dimension n.
For each term z 1 of (p (z) or y (z) is transformed into the term
Z2(z- L lz 2 ) i = * ""**! of dimension w, by clearing of fractions.

We come now to the detailed study of the functional dependence which
our equation (1) or, as the case may be, (3) establishes between z and w.
We shall apply consistently our two new aids, mapping upon the complex
sphere and homogeneous coordinates. We shall have solved this problem
when we can form a complete picture of the conformal relation between
the ^-sphere and the Riemann surface over the z^-sphere.

First of all we must inquire as to the nature and the position of the
branch points of the Riemann surface. I remind you here that a /J-fold
branch point is one in which ^ + 1 leaves are connected. Since w is
a single-valued function of z, we know the branch points when we know
the points of the z sphere which correspond to them, which I am in the
habit of calling the critical or noteworthy points of the z-sphere. To
each of these there corresponds a certain multiplicity equal to that of
the corresponding branch point. I shall now give, without detailed
proof, the theorems which make possible the determination of these
points. I assume that the rather simple functiontheoretic facts which
enter into consideration here are in general familiar to you, though they
may not be in the homogeneous form which I prefer to use. I shall
illustrate in concrete graphical form the abstract considerations which
I shall present to you, in this connection, by a series of examples.

A little calculation is necessary in order to obtain the analogue, in
homogeneous coordinates, of the differential coefficient dw/dz. Differ-
entiating equation (3) and omitting the bars over q> and y, we obtain

_ y>d<p

We have also

dq) =

dip =
where

108

Algebra :" Equations in the Field of Complex Quantities.

On the other hand, from Euler's theorem for homogeneous functions of
degree n, we have

9?i #1 -f- % ' ^2 = w <p

= n

consequently the numerator on the right side of (3') may be written
in the form

\pd(p

d<p, dip
 V

<p l dz l -f

This expression, by the multiplication theorem for determinants, becomes

Thus (3') goes over into the equation

This constitutes the basal formula of the homogeneous theory of our
equation, and the functional determinant (p l y 2 <P% V ; i of the forms cp , ip
appears as a crucial expression for all that follows. Except for it and
for the factor 4/(^ V 2 ) one has on the right the differential of z = z^z^,
on the left that of w = wjw^. Since for finite z and w the critical
points are given by dw/dz = 0, as is well known, the following theorem
appears plausible, but I shall here omit the proof. Each /i-fold zero of
the functional determinant is a critical point of multiplicity //, i.e.,
there corresponds to it a /Li-fold branch point of the Riemann surface over
the w-sphere. The chief advantage of this rule, as compared with those
which are otherwise given, lies in the fact that it contains in one statement
both finite and infinite values of z and w. It enables us also to make
a precise statement concerning the number of remarkable points. The four
derivatives, namely, are forms of dimension n \ , and the functional
determinant is therefore a form of dimension 2 n 2 . Such a polynomial
always has 2 n 2 zeros, if one takes into account their multiplicity.
Thus, if HI, & 2 , . . ., oc v are the remarkable points of the z-sphere (i.e., if

their respective multiplicities, then their sum is

it _i- II J- . . . _L- II O*f ">

jt/j [^ A 2 1^ T^ r"V ~* *

By virtue of the conformal mapping, to these points there correspond
the v branch points

a l , a 2 , . . . , a v

on the Riemann surface over the w-sphere, which must necessarily lie
separated on the surface, and about which fa + 1 , fa + 1 , . . . , /* v + \
leaves, respectively, must be cyclically connected. It should be noted,

Equations with a Complex Parameter. 109

however, that different ones of these branch points may lie over the
same position on the w sphere, since w = <p (z)/y(z) for z = <x lt <x 2 . , . . , <x v
may give the same value for w more than once. Over such a point,
there would be two or more separate series of leaves, each series being
in itself connected. Every such position on the w sphere is called a
branch position; we shall denote them, in order, by A , B , C , . . . . It
should be noted that their number can be smaller than v.

The statements thus far made furnish only a hazy picture of the
Riemann surface. We shall now build it up so that it can be more readily
visualized. For this purpose, let us draw on the w sphere through the branch
positions A , B , C , . . . an arbitrary closed curve ( without double points
and of the simplest possible form (see Fig. 38), and distinguish the two
spherical caps thus formed as the upper cap and
the lower cap. In all of the examples which
we shall discuss later the points A , B , C , . . .
will all be real and we shall then naturally
select as the curve ( the meridian great circle of
real numbers, so that each of our two partial
regions will be a hemisphere.

Returning to the general case we see that
each pair of leaves of the Riemann surface
which are connected, intersect along a branch rig. 33.

cut which joins two branch points. As you

know, the Riemann surface remains unchanged in essence if we move
these cuts, leaving the end points fixed, that is, if we think of the
same leaves as being connected along other curves, provided these
join the same branch points. It is in just this variability that the
great generality and also the great difficulty of the idea of the Rie-
mann surface lies. In order to give the surface a definite form, which
shall be susceptible of concrete visualization, we move all the branch cuts
so that all of them lie over the curve & , which passes through all the branch
points. It may be that several branch cuts lie over the same part of
(, and none at all over other parts.

Now let us cut this entire complex of leaves, i.e., each individual leaf,
along the curve (. Since we had already moved all the branch cuts into
position over (, the incision just made passes along all of them, so that
our Riemann surface separates into 2n "half -leaves" entirely free from
branches, n of them over each of the two spherical caps. If we think of
the half -leaves corresponding to the upper cap as being shaded, and those
corresponding to the lower as not shaded, we can distinguish briefly,
n shaded and n unshaded half -leaves. We can now describe the original
Riemann surface as follows. On it each shaded half-leaf meets only un-
shaded half -leaves, those with which it is connected along segments of the
curve ( lying over A B , B C , . . . ; and, similarly, each unshaded half-leaf

-J-JQ Algebra: Equations in the Field of Complex Quantities.

is connected along such segments of ( only to shaded half -leaves. However,
more than two half -leaves may meet only at a branch point] and in fact
around any [t-fold branch point, fi + 1 shaded half-leaves would alternate
with ft + 1 unshades ones.

Since the mapping by means of our function w (z) of the z sphere
upon the Riemann surface over the w sphere is a one-to-one correspon-
dence, we can immediately transfer to the z sphere the above conditions
of connectivity. Because of continuity, the 2 n half-leaves of the Rie-
mann surface must correspond to 2n connected z regions, which we
may call the shaded and the unshaded half-regions. These will be
separated from one another by the n images of each of the segments
AB, BC , . . . oi the curve ( which the w-valued function z (w) represents
upon the z sphere. Each shaded half-region meets only shaded half-regions
along these image-curves, and each unshaded half-region meets only shaded
ones. It is only in a ft-fold critical point that more than two half-regions
can meet. At such a point (JL + 1 shaded and // + \ unshaded half -regions
come together.

This division of the z sphere into partial regions will help us to follow
in detail the course of the function z (w) for a few simple characteristic
examples. I shall begin with the simplest one possible.

1. The "pure" equation
We shall call the well known equation

(1 ) z n = w

a pure equation. Its solution is given formally by introducing the

w r~
radical z = \w. This gives us no information, however, regarding the

functional relation between z and w . We shall proceed according to the
general plan by introducing the homogeneous variables

-^i i I '
w* ~ z\*

and we shall consider the functional determinant of the numerator and
denominator of the right side

This expression obviously has the (n 1) fold zeros z l = and z 2 = 0,
or (in non-homogeneous form) 2 = and z = oo. These are the only
critical points and they are of total multiplicity 2n 2. By our
general theorem, therefore, the only branch points of the Riemann surface
over the w sphere are at the positions w = and w = <x>. By the equation
w z n these correspond to the two points z = and z = oo . Each
of these two points has the multiplicity n \ , so that n leaves are

The "Pure" Equation.

Ill

W'Sphere:

cyclically connected at each of them. Let us now mark on the w sphere
the meridian of real numbers as the curve and let us cut all the leaves
of the Riemann surface along this meridian, after having appropriately
displaced all of the branch cuts. Of the 2n hemispheres into which the
surface separates we think of those over the rear half
of the w sphere, that is, those which correspond to w
values with positive imaginary parts, as shaded. Upon
the meridian itself, we shall distinguish between the
half meridian of positive real numbers (drawn full in
Fig. 39) and that of the negative real numbers (dotted).
Now we must examine the mappings of this
meridian ( curve upon the z sphere, where they bring
about the characteristic division into half-regions.
Upon the positive half meridian w = r, where r ranges
through positive real values from tooo; for these values we have
by a well known formula of complex numbers,

2kn\
n )'

z =

= Vr cos-

+ isin-

where & = 0, 1, ..., 1.

For the different values of k , this expression gives those n half-meridians
of the z sphere which make with the half-meridian of positive real numbers
the angles 0, 2 n\n, 4^jn t . . ., 2(n l}n\n. Thus these curves corres-

z-Sphere:

pond to the full drawn half of (. On the negative half -meridian of the
w sphere we must set w = r = r e ijl , where again ^ r ^ oo. This
gives

(2k

^ -

, .

+ ism

(2k +

, where = 0, 1, . . .,n 1.

Corresponding to this we have, on the z sphere, those n half-meridians
which have the "longitude" n/n, $n/n, . . . ,2(n \)n/n, which thus bisect
the angles between the others. Accordingly, the z sphere is divided into
2n congruent sectors reaching from the north pole to the south pole, similar

112 Algebra: Equations in the Field of Complex Quantities.

to the natural divisions of an orange. This division is exactly in accord
with the general theory. In particular, it is only at the remarkable
points, the two poles, that more than two half -regions meet. At each
of these points 2n half -regions meet, corresponding to the multipli-
city n \ .

As for the shading of the regions, we need to fix it for one region only.
The remainder are then alternately shaded and unshaded. Now note
that when we look at the shaded half of the w sphere (the rear) from the
point w = 0, the full drawn part of the boundary lies to the left, the
dotted part to the right. Since we are concerned with a conformal map-
ping in which angles are not reversed, each shaded portion of the z sphere,
looked at from the correponding point z , must have the same property
as to position, that is, it must have a full drawn boundary to the left, and
a dotted one to the right. With this we control completely the division
of the z sphere into regions. Moreover, one notices a characteristic
difference in the distribution of the regions upon two z hemispheres,
according as n is even or odd, as can be clearly seen in Figs. 40 and 41
on p. Ill for the first cases n = 3, n = 4. Let me emphasize how
necessary it was to go over to the complex sphere in order to get a full
understanding of the situation. In the complex z plane, one would
have a division into angular sectors by straight lines radiating from
z = 0, and it would not be at all so obvious that z = oo and w oo
have equal significance with z = and w = 0, as critical point
and branch point, respectively.

This furnished us with the essentials for exact knowledge of the
functional relation between z and w. We need now study only the
conformal mapping of each of the 2 n spherical sectors upon one or the
other of the two w hemispheres. But I shall not go into the details here.
This case, as one of the simplest and most obvious illustrations, will
be familiar ground to any one who has had to do with conformal re-
presentation. We shall see later (see p. 131) how to deduce from this
methods for the numerical calculation of z.

Let us, however, settle here the important question as to the mutual
relation among the various congruent regions of the z sphere. Speaking more
exactly, w = z n takes on the same value at a point in each one of the n
shaded regions. Can the corresponding values of z be expressed in terms
of one another? We notice, in fact, that for z' = z (where e is any
one of the nth roots of unity) z' n = z n , that is w = z n takes the same
value at all the n positions

(2)

= g'' . z = e n -z (v = 0, 1, 2, . . . , n 1).

These n values of z' must therefore be distributed so that just one of
them lies in each of the n shaded regions of the z sphere, if z is taken

The "Pure" Equation.

in one of the shaded regions and each of them must traverse one of
these regions as z traverses its region. The same thing is true of the
unshaded regions. Each of the substitutions (2) is represented geo-
metrically by a rotation of the z sphere through an angle v 2 n\n about
the vertical axis 0, oo, since, as is well known, multiplication in the
complex plane by e 2vijT/n denotes a rotation through that angle about
the origin. Thus corresponding points of our spherical regions, as well
as the regions themselves, go over into one another by means of these n rota-
tions about the vertical axis.

If, then, we had determined at the start only one shaded partial
region of the sphere, this remark would have furnished all the similar
partial regions. In this we have made use only of the property of the
substitutions (2) that they transform equation (\) into itself (i.e., z n = w
into z tn = w) and that their number is equal to the degree. In the examples
that follow, we shall always be able to give such linear substitutions
at the outset, and by means of them to simplily the determination of
the division into subregions.

By using the present example I should like to illustrate an important
general notion, namely, the notion of irreducibility for equations which
contain a parameter w rationally. We have already discussed irreduci-
bility of equations with rational numerical coefficients in connection with
the construction of the regular heptagon (p. 51 et seq.). An equation
f (z t w) = (e.g., our equation z n w = 0), where f (z t w) is a poly-
nomial in z , whose coefficients are rational functions of w , is called reducible
with respect to the parameter w , when f can be split into the product of
two polynomials of the same sort, in each of which z really appears

f (z,w) =f l (z,w) f 2 (z,w)',

otherwise the equation is called irreducible with respect to w. The entire
generalization, in comparison with the earlier conception, lies in the
fact that the tf domain of rationality" in which we operate and in which
the coefficients of the admissible polynomials are to lie, consists of the
totality of rational functions of the parameter w instead of the totality of
rational numbers, in other words, that we pass from a numbertheoretic
to a functiontheoretic conception.

If we illustrate this, for each equation / (z, w} = 0, by means of its
Riemann surface, we can set up a simple criterion for reducibility in this
new sense. If the equation, namely, is reducible, every system of the
values z, w which satisfies it satisfies either f (z, w} = or / 2 (z, w) = 0;
now the solutions of / t = and / 2 = are represented by means of
their Riemann surfaces, which have nothing to do with each other,
and, in particular, are not connected. Thus, the Riemann surface which
belongs to a reducible equation f (z, w) = must break down into at least
two separates pieces.

Algebra: Equations in the Field of Complex Quantities.

According to this, we can now assert that the equation z n w =
is certainly irreducible in the function theoretic sense. For, on its Riemann
surface, which we known exactly, all the n leaves are cyclically connected
at each of its branch points. Moreover, the entire surface is mapped upon
the unpartitioned z sphere. Hence such a breaking down cannot occur.
In connection with this, we can answer one of the popular problems of
mathematics which we touched earlier (p. 51), namely, that of the possibility
of dividing an arbitrary angle <p into n equal parts, in particular, for n = 3,
the possibility of trisecting an angle. The problem is to give an exact
construction with ruler and compasses for dividing into three equal parts
any angle (p whatever. (It is easy, of course, to give a construction for
a series of special values of <p). I shall give you the train of thought
for the proof of the impossibility of trisecting an angle in the sense just

mentioned, and I shall ask you to recall, in
this connection, the proof of the impossibility
of constructing the regular heptagon with
ruler and compasses (see p. 51 et seq.). Just
as at that time, we shall reduce the problem
to that of the solution of an irreducible cubic
equation, and we shall then show that this
equation cannot be solved by a series of
Fig. 42. square roots; except that, now, the equation

will contain a parameter (the angle (p) , whereas,

before, the coefficients were integers. Accordingly, functiontheoretic
irreducibility must replace numbertheoretic irreducibility.

In order to set up the equation of the problem let us think of the
angle (p as laid off from the positive real half -axis in the w plane (see Fig. 42) .
Then its free arm will cut the unit circle in the point

w = e i( v = cos9? + isincp.

Our problem consists in finding, independently of special values of the
parameter <p, a construction, involving a finite number of applications
of the ruler and compasses, which shall give the point of intersection
with the unit circle of the arm of the angle 90/3 , i. e., the point

tv_
z = e 3 = cos Y + isin Y

This value of z satisfies the equation:

(3) z* = cosy + isiiKp ,

and the analytic equivalent of our geometric problem consists in solving
this equation (see p. 51) by means of a finite number of square roots,
one over another, of rational functions of sin (p and cos 9? , since these
quantities are the coordinates of the point w with which we start the
construction

The Dihedral Equation.

We must show, first, that the equation (3) is irreducible in the function
theoretic sense. To be sure, this equation does not have just the form
we assumed while explaining the notion, since, instead of the a complex
parameter w that enters rationally, we have now two functions cos
and sin of a real parameter <p, both of which appear rationally. As a
natural extension here of our notion, we shall call the polynomial
z 3 (cos ) reducible if it can be split into polynomials whose
coefficients are likewise rational finctions of co .; (p and sin (p ; and we
can, as before, assign a criterion for this. If we let (p assume all real
values in (3), w = e itp = cos 9? + i sin (p will describe the unit circle of
the w plane, to which the equation of the w sphere corresponds by stereo-
graphic projection. The curve which lies over this, on the Riemann
surface of the equation 3 = w, and which describes, in one stroke,
all three leaves, is mapped by equation (3) uniquely upon the unit
circle of the z sphere. Hence it can be regarded, in a sense as its "one
dimensional Riemann image". In the same way, we can obviously
assign such a Riemann image to every equation of the form / (z , cos q> ,
sin <p) = by taking as many copies of the unit circle with arc length (p
as the equation has roots, and joining them according to the connectivity
of the roots. It follows, just as before, that the equation (3) can be reducible
only when its one-dimensional Riemann image breaks down into separate
parts, and this is obviously not the case. This proves the function theoretic
irreducibility of our equation (3).

Now, however, the former proof of the theorem, that a cubic equation
with rational numerical coefficients is reducible if it can be solved by
a series of square roots, can be applied literally to the present case of
the function-theoretically irreducible equation (3) (see p. 51 et seq.).
We need only to replace "rational numbers" there by "rational functions
of cos <p and sin <p" . This proves our assertion that the trisection of an
arbitrary angle cannot be accomplished by a finite number of applications
of a ruler and compasses. Hence the endeavors of angle -trisection
zealots must always be fruitless!

I pass on now to the treatment of a somewhat more complicated
example.

2. The dihedral equation
The equation

is called the dihedral equation, for reasons that will appear later.
Clearing of fractions, we see that its degree is 2n. Introducing homo-
geneous variables we get

116 Algebra: Equations in the Field of Complex Quantities.

in which, in fact, forms of dimension 2n appear in numerator and
denominator. The functional determinant of these forms is

It has an (n l)-fold zero at 2 X = and at z a = 0; the other 2n zeros
are given by

4-zl = or:
If in addition to the n-th root of unity

2tw_

E = e n

which we have already used, we introduce also the primitive w-th root
of -1:

the last 2n zeros are given by the equations

^ = e" and - 1 - = e- e v , (v = 0, 1 , . . . , n 1) .

^2 Z 2,

Since the values of z = ^/^ corresponding to them all have the absolute
value one, they all lie therefore on the equator of the z sphere (corres-
ponding to the unit circle of the z plane), at equal angular spacings of n\n .
We have therefore as critical points on the z sphere:

(a) the south pole z = and the north pole z = oo , each of multiplicity
n-\\

(b) the 2n equatorial points z = e r , f' e v , each of multiplicity one.
The sum of all the multiplicities is 2 (n 1) + 2n \ = 4n 2,
as is demanded by the general theorem on p. 108 for the degree 2n.
By virtue of equation (1) there will correspond to the remarkable points
z = o, z = oo of the z sphere, the position w = oo on the w sphere.
Moreover, to all the points z = r , corresponds the position w = +1 ;
and, to all the points z = e t? the position w = \ . There are, accord-
ingly, only three branch points <x>, +1, \ on the w sphere. These
will lie as follows:

w = oo two branch points of multiplicity n \ ;
w = +1 branch points of multiplicity 1;
w = i w branch points of multiplicity 1.

TA0 2w Ztfflfltfs o/ /As Riemann surface group themselves therefore over
the point w = oo in two separate series, each of n cyclically connected
leaves', over w = +1 and w = 1 iw n series, each of two leaves. The
disposition of the leaves will become clear when we study the corres-
ponding subdivision of the z sphere into half-regions.

The Dihedral Equation.

117

To this end it will be well, as we remarked above, to know the linear
substitutions which transform equation (1) into itself. As in the case of
the pure equation, it is unchanged by the n substitutions

\
( 23 -)

n \), where e = e

since for these z' n = z n . Likewise, however, it is unchanged by the n
additional substitutions

(2b

z' = ~(v = Q,\, ...n-\).

since these only change z n into \/z n .

We have therefore 2n linear substitutions of equation (1) into itself,
exactly as many as its degree indicates. Thus, if we know for a given
value W Q of w one root Z Q of the equation, we know immediately 2 n roots

w-Sphere:

^- Sphere:

v Z Q and e v /z (v = , 1 , 2 , . . . , n 1 ) , in general all different, for which w
has the same value W Q , i. e., we know all the roots of the equation when
we have obtained the n-th root of unity e .

Let us now proceed to examine the subdivision of the z sphere corres-
ponding to cuts along the real meridian of the Riemann surface over the
w sphere. In this, as in the previous example,we distinguish on the real
meridian of the w sphere the three segments made by the branch points
that from +1 to oo (drawn full), that from oo to 1 (short dotted),
and that from 1 to +1 (long dotted) (see Fig. 43). To each of these
three segments there correspond on the z sphere 2n different curvilinear
segments which can be derived from any one of them by means of the 2 n
linear substitutions (2). It will always suffice, therefore, to find one of
them. Moreover all these segments must connect the critical points
z = o, oo, e v , e' e v , which we therefore mark on the z sphere. Just as
in the previous case, their form is of a somewhat different type according
as n is even or odd. It will suffice if we exhibit a definite case, say for
n = 6. Fig. 43 shows the front half of the z sphere in orthogonal pro-
jection. One sees, on the equator, from left to right with spacings of

H8 Algebra: Equations in the Field of Complex Quantities.

60, 3 = 1 , 4 , 5 , e 6 = 1 ; and lying midway between the others, e 7 - e 3 ,
E' 4 = i , and e' 5 .

A/oze> w shall see that the quadrant +1 < 2 < oo of the meridian of
real z corresponds to the part of the real w meridian +1 <w<oo (full
drawn). In fact, if we put z = r and let r range through real values
from 1 to oo, then w = i (z n +\jz n } = \ (r n + \jr n ) will vary also through
real values that are always increasing, from 1 to oo . We obtain n other
full drawn curves on the z sphere, from this one, by means of the n linear
substitutions (2 a). But, as we saw in the previous example, these
substitutions mean rotations of the sphere about the vertical axis (0 , oo)
through the angles 2njn, 4n/n, . . . , 2 (n 1) n\n. We get in this way
the n quarter-meridians from the north pole oo to the points r on the
equator. We get an additional full drawn curve if we apply
the substitution z' = \\z, which transforms the meridian quadrant
from +1 to oo into the lower real meridianquadrant from
+ 1 to 0. If we subject this quadrant to the n rotations (2 a),
the composition of which with z' = \jz gives the n substitutions
(2b), we obtain, in addition, the n meridian quadrants which join the
south pole with the equatorial points e v . We have now in fact the 2n
full drawn curves which correspond to the full drawn w meridian qua-
drant. In particular, for n = 6, they make up the three entire meridians
into which the real meridian is transformed by rotations of 0, 60, 120.

It is now also obvious that the totality of the values z = e' r,
where r again ranges through real values from -\-i to oo, corresponds
to the dotted part of the real w meridian; for the equation (1) yields then:

_ 1 / 'n w I 1 \ __ 1 /n M

2 \ ' / n /yii i 2 \ ' /y n I *

and this expression actually decreases through real values from 1 to
oo. But z = e' r represents the meridian quadrant from oc to the
equatorial point e v . If we now apply to it the substitutions (2 a), (2b),
we find, as before, that to the dotted part of the real w meridian there corres-
pond all the meridian quadrants joining the poles to the equatorial points
e e v , which thus bisect the angles between the meridian quadrants which
we obtained before. In particular, for n = 6, they make up the three
entire meridians into which the real meridian is transformed by
rotations of 30, 90, 150.

There remain to be found the 2n curvilinear segments which corres-
pond to the long-dotted half-meridian 1<10<+1. I shall prove
that they are the segments of the equator of the z sphere determined by
the points e v and e' e v . In fact, the equator represents the points of
absolute value one and is given therefore by z = e i<p where (p is real
and ranges from to 2n. Hence we have

The Dihedral Equation. \\g

This expression is always real, and its absolute value is not greater than 1 ,
In fact, it assumes once every value between + 1 and 1 as <p varies
from one multiple of n\n to the next one, i.e., when z traverses one of
the segments of which we are speaking.

The curves determined in this manner divide the z sphere into 2 2 n
triangular half-regions which are bounded by one curve of each of the three
sorts, and each half-region corresponds to a half leaf of the Riemann surface.
Several regions can meet only at the critical points, and then in accord-
ance with the table of multiplicities (p. 116), namely, 2n at the north
pole, and at the south pole, and 2 2 at each of the points e 1 ' and e' e v .
In order to determine which of these regions are to be shaded, we notice
that when w traverses, in order/ the full-drwan, the long-dotted, and
the short dotted parts of the real w meridian, the rear half of the w sphere
lies at its left. Since the mapping is conformal with preservation of
angles, we should shade those half-regions whose boundaries follow
one another in this same sense, and we should leave the others unshaded.

We have now obtained a complete geometric picture of the mutual
dependence between z and w which is set up by our equation. We might
follow it out in greater detail by Examining more closely the conformal
mapping of the single triangular regions upon the w hemisphere, but we
shall forego this. / shall describe only, and briefly, the case n 6,to which
I have already given special attention. The z sphere is then divided into
twelve shaded and twelve unshaded triangles of which six of each sort
are visible in Fig. 44. Six of each sort meet at each pole, and two of
each sort at each of twelve equidistant points of the equator. Each
triangle is mapped conformally upon a w half -leaf of the same sort. Of
the half-leaves of the Riemann surface, six of each sort are connected
at the branch position oo , and two of each sort at each of the branch
positions ^ 1 > corresponding to the grouping of the half-regions on
the z sphere.

We may obtain a convenient picture of the division of the z sphere,
and one which is especially valuable because of its analogy with pictures
soon to come, as follows. If we join the n equidistant points on the
equator (e. g., the e") with one another in order by straight lines,
and also join each of them to the two poles, one obtains a double pyramid,
with 2n faces, inscribed in the sphere (in Fig. 44, twelve faces). If we
now project, from the center, the subdivision of the z sphere upon this
double pyramid, every pyramid face is divided into a shaded and an
unshaded half by the altitude of that face dropped from the pole. If
we represent the division of the z sphere, and consequently our function,
by means of this double pyramid, the latter will render a service quite
analogous to that which we shall get in the coming examples from the
regular polyhedra. We obtain a complete analogy if we think of the double
pyramid as collapsed into its base, and consider the double regular n-gon

120

Algebra: Equations in the Field of Complex Quantities.

(hexagon) which results whose two faces (upper and lower) are divided
each into 2 n triangles by the straight lines which join the center with
the vertices and the middle points of the sides (see Fig. 45). / have
been in the habit of calling this figure a dihedron and of classing it with
the five regular polyhedra which have been studied since Plato's time.
It fulfills, in fact, all the conditions by means of which a regular poly-
hedron is usually defined, since its faces (the two faces of the w-gon)
are congruent regular polygons, and since it has congruent edges (the
sides of the n-gon) and congruent vertices (the vertices of the n-gon).
The only difference is that it does not bound a proper solid body but
encloses the volume zero. Thus the theorem of Plato, that there are

Fig. 44.

Fig. 45.

only five regular solids, is correct only when one includes in the definition
the requirement of a proper solid, which is always tacitly assumed in
the proof.

// we start with the dihedron, we obtain our subdivision of the z sphere
by projecting upon that sphere not only its vertices but also the centers of
its edges and its faces, the projecting rays for the latter being perpendi-
cular to the plane of the dihedron. Thus the dihedron can also be looked
upon as representing the functional relation which our equation sets up
between w and z. Hence the brief name which we have already used,
dihedral equation, is appropriate.

In addition, we shall now consider those equations which, as already
intimated, are closely related to the platonic regular solids.

3. The tetrahedral, the octahedral, and the icosahedral equations.
We shall see that the last two could, with equal right, be called the
hexahedral and the dodecahedral equations, so that all five regular
bodies will have been covered. We shall follow here a route that is
the reverse of the one we followed in the preceding example. Starting
from the regular body, we shall first deduce a division of the sphere into
regions, and we shall then set up the appropriate algebraic equation, for
which that figure is the proper geometric interpretation. I shall have to
confine myself frequently to suggestions, however, and I therefore refer
you at once to my book: Vorlesungen ilber das Ikosaeder und die Auf-

The Tetrahedral, the Octahedral, and the Icosahedral Equations. \2\

losung der Gleichungen vom funften Grade 1 , in which you will find a
systematic presentation of the entire extensive theory with its numerous
relations to allied fields.

Moreover, I shall give a parallel treatment of all three cases and
I shall begin by deducing the subdivision of the sphere for the tetrahedron.

\. The tetrahedron (see Fig. 46). We divide each of the four equi-
lateral face-triangles of the tetrahdron, by means of the three altitudes,
into six partial triangles.
These are congruent in
two groups of three each,
while any two non-
congruent ones are sym-
metric. We obtain thus
a division of the entire
surface of the tetrahedron
into twenty -four triangles,
which fall into two groups,
each containing twelve

Face Triangle (actual size).

Tetrahedron.

congruent triangles, while Fig. 46.

any triangle of one group

is symmetric to every triangle of the other group. We shall shade the

triangles of one group. Among the vertices of these twenty-four

triangles we can distinguish three sorts, such that each triangle has one

vertex of each sort:

a) the four vertices of the initial tetrahedron, at each of which three
shaded and three unshaded triangles meet',

b) the four centers of gravity of the faces, which determine again
another regular tetrahedron (the co-tetrahedron) ; at each of these, three
triangles of each kind meet',

c) the six middle points of the edges, which determine a regular octa-
hedron', at each of these, two triangles of each kind meet.

If from the center of gravity of the tetrahedron we project this subdivision
into triangles upon the circumscribed sphere, the latter will be subdivided
into 2 12 triangles, which are bounded by arcs of great circles and are
mutually congruent or symmetric. About each vertex of the sort a), b), c),
there will be respectively 6, 6, 4 equal angles, and since the sum of the
angles about a point on a sphere is 2^, each of the spherical triangles
will have an angle rc/3 at a vertex of the sort a or b and an angle n/2 at a
vertex of the sort c.

It is a characteristic property of this division of the sphere that it,
as well as the tetrahedron itself, is transformed into itself by a number

1 Leipzig 1884; referred to hereafter as "Ikosaeder". Translation into English
by G. C. Morrice: Lectures on the Icosahedron by Klein. Revised Edition, 1911,
Kegan Paul & Co.

122

Algebra: Equations in the Field of Complex Quantities.

of rotations of the sphere about its center. This will be clear to you in
detail if you examine a model of the tetrahedron with its divisions,
like the one in our collection. For the lecture, it will suffice if I indicate
the number of possible rotations (whereby the position of rest is included
as the identical rotation. If we select a definite vertex of the original
tetrahedron, we can, by means of a rotation, transform it into every
vertex of the tetrahedron (including itself), which gives four possibilities.
If we keep this vertex fixed, however, in any one of these four positions,
we can still transform the tetrahedron. This gives altogether 4 3 = 12
rotations which transform the tetrahedron, or the corresponding tri-
angular division of the circumscribed sphere, into itself. By means of
these rotations we can transform a preassigned shaded (or unshaded)
triangle into every other shaded (or unshaded) triangle, and the particular
rotation is determined when that second triangle is chosen. These
twelve rotations form obviously what one calls a group G 12 of twelve
operations, i.e., if we performs two of them in succession, the result
is one of the twelve rotations.

If we think of this sphere as the z sphere, each of these twelve
rotations will be represented by a linear transformations of z, and the

twelve linear transformations which
arise in this manner will transform
into itself the equation which cor-
responds to the tetrahedron. For pur-
poses of comparison, I remark that
one can interpret the 2 n linear sub-
stitutions of the dihedral equation as
the totality of the rotations of the
dihedron into itself.

2. We shall now treat the octa-
hedron similarly (see Fig. 47) and
we may be somewhat briefer. We
divide each of the faces, just as
before, into six partial triangles and
obtain a division of the entire surface of the octahedron into twenty-four
congruent shaded triangles, and twenty-four unshaded triangles which are
congruent among themselves but symmetric to the other twenty-four. We
can again distinguish three sorts of vertices:

a) the six vertices of the octahedron, at each of which four triangles
of each kind meet\

b) the eight centers of gravity of the faces, which form the vertices of
a cube\ at each of these, three triangles of each kind meet;

c) the twelve mid-points of the edges, at each of which two triangles
of each kind meet.

The Tetrahedral, the Octahedral, and the Icosahedral Equations. -123

If we pass now to the circumscribed sphere, by means of central pro-
jection, we obtain a division into 2 24 spherical triangles which are
either congruent or symmetric, and each of which has an angle rc/4 at
the vertex a , nfy at the vertex b , and n/2 at the vertex c . Since the
vertices b form a cube, it is easy to see that one would have obtained the
same division on the sphere if one had started with a cube and had projected
its vertices, and the centers of its faces and edges, upon the sphere. In other
words, we do not need to give special attention to the cube.

Just as in the previous case, it is easy to see that the octahedron,
as well as this division of
the sphere t is transformed
into itself by twenty-four
rotations which form a group
G 24 ; again each rotation is
determined in that it trans-
forms a preassigned shaded
triangle into another definite
shaded triangle.

3. We come finally to
the icosahedron (see Fig. 48) .
Here, also, we start with
the same subdivision of
each of the twenty-four
triangular faces and obtain
altogether sixty shaded and
sixty unshaded partial tri-
angles. The three sorts of pig. 48.
vertices are:

a) the twelve vertices of the icosahedron, at each of which five triangles
of each kind meet]

b) the twenty centers of gravity of the faces, which are the vertices of a
regular dodecahedron', at each of them three triangles of each kind meet]

c) the thirty mid-points of the edges, at each of which two triangles of
each sort meet.

When this is carried over to the sphere each spherical triangle has
at the vertices a t b t c the angles rc/5, nr/3, rc/2, respectively. From the
property of the vertices b one can conclude, as before, that the same
division of the sphere would have resulted if one had considered the dodeca-
hedron.

Finally, the icosahedron, as well as the corresponding division of the
sphere, is transformed into itself by a group G 60 of sixty rotations of the
sphere about its center. These rotations, as well as those for the octa-
hedron, will become clear to you upon examination of a model.

124 Algebra: Equations in the Field of Complex Quantities.

Let me make a list of the angles of the spherical triangles which have
appeared in the three cases which we have considered, to which I shall
add the dihedron also; they are

Dihedron : 7i/2 , n\2 , n\n ;

Tetrahedron: aft, jr/3 , rc/2;

Octahedron: n/4, n/3, n\2\

Icosahedron: jr/5, ft/3, n/2.

As a variation of a joke of Kummer's I might suggest that the
student of natural science would at once conclude from this, that
there were additional subdivisions of the sphere, having analogous
properties, and with angles such as Ji/6, nft, n\2\ n\7 , nft , 71/2. The
mathematician, to be sure, does not risk making such inferences by
analogy, and his cautiousness justifies itself here, for the series of possible
spherical subdivisions of this sort ends, in fact, with our list. Of course
this is connected with the fact that there are no more regular polyhedrons.
We can see the ultimate reason in a property of whole numbers, which
does not admit a reduction to simpler reasons. It appears, namely,
that the angles of each of our triangles must be aliquot parts of n t
say n/m, yi/n, n\r , such that the denominators satisfy the inequality

\\m + \\n + \\r > 1 .

This inequality has the property of existing only for the integral solutions
given above. Moreover, we can understand it readily, since it only
expresses the fact that the sum of the angles of a spherical triangle
exceeds n.

I should like to mention that, as some of you doubtless know, an
appropriate generalization of the theory does carry one byeond these
apparently too narrow bounds: The theory of automorphic functions in-
volves subdividing the sphere into infinitely many triangles whose angle
sum is less than or equal to n.

4. Continuation: Setting up the Normal Equation.
We come now to the second part of our problem, to set up that
equation of the form

(1) V (z)-u,v(,) = 0, or = ,

which belongs to a definite one of our three spherical subdivisions, that is,
which maps the two hemispheres of the w sphere upon the 2-12, or
the 2 24, or the 2 60 partial triangles of the z sphere. To each value
of w there must correspond then, in general, 12, 24, 60 values, respectively,
of z, each one in a partial triangle of the right kind. Hence the desired
equation must have the degree 12, 24, 60 in the three cases respectively,
for which we shall write N in general. Now each partial region touches

Continuation: Setting up the Normal Equation.

125

w-Sphere:

three critical points; hence there must be, in every case, three branch
positions on the w sphere. We assign these, as is customary, to w = 0,
1 , oo i and we choose again the meridian of real numbers as the section
curve ( through these three points, whose three segments shall correspond
to the boundaries of the z triangles.

We shall assume (see Fig. 49) that in each of the three cases the
centers of gravity of the faces (vertices b in the former notation) correspond
to the point w = , the mid-point of the edges (vertices c) to the point w = \ ,
and the vertices of the polyhedron (vertices a) to the point w = <x>. The sides
of the triangles will then correspond to the three segments of the w meri-
dian in the manner indicated by the mapping, and the shaded triangles
will correspond to the rear w hemisphere, the unshaded to the front w
hemisphere. By virtue of these correspon-
dences, the equation (1) is to effect a unique
mapping of the z sphere upon an JV-leaved
Riemann surface over the w sphere with
branch points at , 1 , o .

We might deduce, a priori, a proof for the
existence of this equation by means of general
functiontheoretic theorems. However, I prefer
not to presuppose the knowledge which this
method would require, but to construct the
various equations empirically. This method
will give us perhaps a more vivid perception
of the individual cases.

Let us think of equation (1) written in
homogeneous variables

rv-

7V*7

Fig. 49-

where <& N , 1 F N are homogeneous polynomials of dimension N in z lt z z
(N = 12, 24, or 60). In this form of the equation, the positions w^ = 0,
w 2 = (i.e., w = 0, oo) on the w sphere seem to be favored more than
the third branch position w = 1 (in homogeneous form, w^ w 2 = 0).
Since, however, the three branch positions are, for our purpose, of equal
importance, it is expedient to consider also the following form of the
equation :

where X N = <&N Vy denotes also a form of dimension N. Both forms
are embraced in the continued proportion

(2) v>i : (w l w 2 ) :w 2 = $ N (z l , z 2 ) : X N (z l , * a ) : N (z l ,z 2 ).

This furnishes us with a completely homogeneous form of equation (1)
which gives the same consideration to all the branch points.

126 Algebra: Equations in the Field of Complex Quantities.

Our problem now is to set up the forms <&N , XN , Y N . For this purpose,
we shall bring them into relation to our subdivision of the z sphere.
From equation (2) we see that the form 0y (z lt z 2 ) = for w l = 0, i. e.,
that w = corresponds to the N zeros of &N on the z sphere. On the other
hand, the centers of gravity of the faces of the polyhedron (vertices b in the
subdivision), of which there are JV/3 in every case, must, according to
our assumptions, correspond to the branch position w = 0. But every
one of these centers must be a triple root of our equation, since in each
of them there meet three shaded and three unshaded triangles of the
z sphere. Thus these points, each with multiplicity three, supply all the
positions which correspond to w = 0, and consequently all the zeros of
&y. Hence <P N has only triple zeros and must, therefore, be the third
power of a form <pn (z l , z 2 ) of degree Af/3 :

In the same way, it follows that the zeros of XN = correspond to
the position w = \ (i. e., w l w 2 = 0), and that these are identical with
the N/2 midpoints, each counted twice, of the edges of the polyhedron
(vertices c of our subdivision). Consequently XN must be the square
of a form of dimension N/2:

Finally the zeros of *P N are to correspond to the point 10 = 00, so that
they must be identical with the vertices of the polyhedron (vertices a
of the subdivision); but at these vertices 3, 4, or 5 triangles meet, in
the several cases, so that we get

YN = bp N!v (*i , * 2 )] v . where v = 3, 4 or 5 .
Our equation (2) must then necessarily have the form

(3) MI : K - w a ) w* = v(*i' Z 2? :*(*i. *a) a : V(*i> *a)">

where the degrees and powers of (p, %, y, and the values of the degree N
of the equation are exhibited in the following table:

Tetrahedron: <pl , jfa , yl ; N = 12.
Octahedron: 9$ , % 2 12 , yg ; N = 24.
Icosahedron : <p!! , xlo > v4 ; N = 60.

I shall now show briefly that the dihedral equation which we discussed,
fits also into the scheme (3). We need only to recall that in that case
we chose 1 , +1, oo as the branch positions on the w sphere instead
of 0, +1, oc which we selected later. We shall, then, obtain actual
analogy with (3) only if we throw the dihedral equation into the form

(u>i + w 2 ) :(w l w 2 ):w 2 =&:X: W.

Continuation: Setting up the Normal Equation.
Now from the dihedral equation (p. 115) which we used:

127

we get by simple reduction

( Wl + w 2 ) : K- ^ 2 ) : w 2 = (z\

2*1*3) :

Thus we can, in fact, add to the above table:

Dihedron : <pl, %*, yl\ N = 2n.

The critical points together with their multiplicities which can at
once be read off from this form of the equation are in full agreement
with those which we found above (see p. 116).

We come now to the actual setting up of the forms <p, %, y in the
three new cases. I shall give details here only for the octahedron, for which
the relations turn out to be the simplest.
But even here I shall, at times, give only
suggestions or results, in order to remain
within the confines of a brief survey. For
those who desire more, there is easily
accessible the detailed exposition in my
book on the icosahedron. For the sake of
simplicity we think of the octahedron as
so inscribed in the z sphere that the six
vertices fall on (see Fig. 50) :

z = 0, oo, + 1, +i, 1, - i.

It will then be a simple matter to give the twenty-jour linear substitutions
of z which represent the rotations of the octahedron, i.e., which permute
these six points. We begin with the four rotations in which the vertices
and oo remain fixed

(4a) z' = i k -z, (6 = 0,1,2,3).

Then we can interchange the points 0, oo by means of the substitution
z' = \\z (i. e., a rotation through 180 about the horizontal axis (+1 , 1)
which transforms every point of the octahedron into another one. If
we now apply the four rotations (4 a), we get four new substitutions:

(4b)

I K
z

(k = 0, 1, 2, 3)

In the same way, we now throw in succession the four remaining vertices
z = \ , i t 1 , i to oo by means of the substitutions

z =

z--\

z \

Z I'

7+1'

which obviously permute the six vertices of the octahedron, and again

128 Algebra: Equations in the Field of Complex Quantities.

apply, each time, the four rotations (4 a). Thus we get 4-4 = 16 ad-
ditional substitutions for the octahedron

(4c)

z i ' z + i '

We have therefore found the desired twenty-four substitutions, and
we can easily show, by calculation, that they really permute the six vertices
of the octahedron and that they form a group G 24 , i. e., that the successive
application of any two of them gives again one of the substitutions in (4).

I shall now construct the form ^ 6 which vanishes in each of the
vertices of the octahedron. The point z = gives the factor z lf the
point z = oo the factor z 2 ; the form z\ z\ has a simple zero at each
of the points 1 , i, so that we obtain finally

(5a) ' y> 6 = z l z 2 (z\ z .

It is more difficult to construct the forms <p s and # 12 which have
as zeros the centers of gravity of the faces and the midpoints of the
edges. Without deducing them, I may state that they are 1

7>8 ^H

It goes without saying that there is an undetermined constant
multiplier in each of these three forms. If g? 8 , y e> # 12 stand for the
normal forms (5), we must insert, in the octahedral equation (3), two
undetermined constants c lt c 2 , and we must write

w l : (w l - w 2 ) :w 2 = <p% : c, j& : c a yj.

The constants c are now to be so determined that these two equations
give actually only one equation between z and w . This is possible when
and only when

is an identity in z l and z 2 . Now this relation can be satisfied by definite
constants c x and c 2 . A brief calculation shows that the identity

must hold, so that the octahedral equation (3) becomes:
(6) w l : (w l - w z ) : w 2 = $ : j& : 108 yj -

This equation surely maps the points , 1 , oo respectively upon the
centers of gravity of the faces, the midpoints of the edges, and the vertices
of the octahedron, with the proper multiplicity, because the forms 90, %, y
were so constructed. Furthermore, the twenty-four octahedron substi-

1 See Ihosaeder, p. 54.

Continuation: Setting up the Normal Equation.

tutions (4) transform it into itself, for they transform the zeros of each
of the forms <p,%, y into themselves and at the same time change
each of the forms by a multiplicative factor. And calculation shows
that these factors cancel when the quotients are formed.

It only remains to show that equation (6) really maps each shaded or
unshaded triangle of the z sphere conformally upon the rear or front w hemi-
sphere. We know that the points 0, 1 , oo of the real w meridian corres-
pond to the three vertuces of each of the triangles; but the equation
has, moreover, twenty-four roots z for each value of w. Since these
must distribute themselves among the twenty-four triangles, w can
take a given value but once, at most, within a triangle. If we could
only show that w remains real on the three sides of a triangle, we could
then easily show that there is a one-to-one mapping of each side upon
a segment of the real w meridian, and also a similar mapping of the
entire interior of the triangle upon the corresponding hemisphere, one which
is conformal without reversal of angles. You will be able to make these
deductions yourselves by making use of the continuity and the analytic
character of the function w (z) . I shall indicate the only noteworthy step
of the proof, that of showing the reality of w upon the sides of the triangle.

It is more convenient to prove this by showing that w is real
upon all the great circles that arise in the octahedral subdivision. These
are, first, the three mutually perpendicular circles which pass each
through four of the six vertices of the octahedron (principal circles]
full drawn in Fig. 50, p. 127), and, second, the six circles, corresponding
to the altitudes of the faces, which bisect the angles of the principal
circles (auxiliary circles] long dotted in Fig. 50). By means of the octa-
hedron substitutions, one can transform every principal circle into any
other and every auxiliary circle into any other. Hence it will suffice
to show that the function w is real at every point on one principal and
one auxiliary circle, since it must take the same values on the other
circles. Now the meridian of real numbers z is one of the principal
circles. By (6), the values on this circle are

which are, of course, real, since y and y are real polynomials in z l and z 2 .
Of the auxiliary circles let us select the one through and oo which makes
an angle of 45 with the real meridian and on which z takes the values

z = e 4 r , where r ranges through real values from oo to + -
On this circle 2 4 = e ia r 4 = r 4 is real. Since by (5) only the fourth
powers of z l and z 2 occur in (p Q and in the fourth power of y e , the last
formula shows that w is real.

This concludes the proof: Equation (6), in fact, maps the w hemisphere,
or the Riemann surface over it, conformally upon that triangular subdivision

-JIQ Algebra: Equations in the Field of Complex Quantities.

of the z sphere which corresponds to the octahedron, and consequently we
have in this case, as completely as in the earlier examples, a geometric
control of the dependence which this equation sets up between z and w.
The treatment of the tetrahedron and of the icosahedron proceeds
according to the same plan. I shall give only the results. As before,
these results are those obtained when the subdivision of the z sphere
has the simplest possible position. The tetrahedral equation 1 is

wi = K ~ ^2)^2 = fe - 2|^3*f*i + 4 s

and the icosahedral equation* is

w, : (w, - w 2 ) : z*> 2 - {- (zf + 2?) + 228 &V 2 - ****) - 494} 3
: -{ (*?+*?) + 522 (zf^-^zf) - 10005 (?

i.e., these equations map the w hemispheres conformally upon the shaded
and the unshaded triangles of that subdivision of the z sphere which belongs
to the tetrahedron and to the icosahedron respectively.

5. Concerning the Solution of the Normal Equations
Let us now consider somewhat the common properties of the equations
which we have been discussing and which we shall call the normal
equations.

Note, first of all, that the extremely simple nature of all our normal
equations is due to the fact that they have exactly the same number of
linear substitutions into themselves as is indicated by the degree, i.e., that
all their roots are linear functions of a single one\ and, further, that we
have, in the divisions of the sphere, a very obvious geometric picture of all
w-Sphere: ^ e re ^ a ^ ons ^ a ^ comeup for consideration. Just how

simple many things appear which are ordinarily
quite complicated with equations of such high degree
will be evident if I raise a certain question in con-
nection with the icosahedral equation.

Let a real value W Q be given, say on the segment
(1 , oo) of the real w meridian (see Fig. 51). Let us
inquire about the sixty roots z of the icosahedral
equation when w W Q . Our theory of the mapping
tells us at once that one of them must lie on a side of each of the sixty
triangles on the z sphere which arise in the case of the icosahedron (drawn
full in Fig. 49, P- 125). This supplies what one calls, in the theory of

1 See Ikosaeder, p. 51, 60. 2 Loc. cit., p. 56, 60.

Concerning the Solution of the Normal Equations.

equations, the separation of the roots, usually a laborious task, which
must precede the numerical calculation of the roots. The task is that of
assigning separated intervals in each of which but one root lies. But we
can also tell at once how many of the roots are real. If we take into
account, namely, that the form of the icosahedral equation given above
implies such a placing 1 of the icosahedron in the z sphere that the real
meridian contains four vertices of each of the three sorts a,b,c, then it
follows (see Fig. 48, p. 123, and Fig. 49, p. 119) that four full-drawn
triangle sides lie on the real meridian, so that there are just four real
roots. The same is true if w lies in one of the other two segments of the
real w meridian, so that for every real w different from , 1 , <x> the icosa-
hedral equation has four real and fifty-six imaginary roots] for w = 0,
1 , oo there are also four different real roots, but they are repeated.

I shall now say something about the actual numerical calculation of
the roots of our normal equations. We have here again the great ad-
vantage that we need to calculate but one root, because the others follow
by linear substitutions. Let me remind you, however, that the numerical
calculation of a root is really a problem of analysis, not of algebra, since
it requires necessarily the application of infinite processes when the root
to which one is approximating is irrational, as is the case in general.

I shall go into details only for the simplest example of all, the pure
equation

w == z rt .

Here I come again into immediate touch with school mathematics. For this

n i

equation, i. e., the calculation of yw, at least for the small values of n
and for real values of w = r, is treated there also. The method of cal-
culating square and cube root, as you learned it in school, depends,
in essence, upon the following procedure. One determines the position
which the radicand w = r has in the series of the squares or cubes,
respectively, of the natural numbers 1, 2, 3,... Then, using the
decimal notation, one makes the same trial with the tenths of the
interval concerned, then with the hundreths, and so on.. In this way
one can, of course, approximate with any desired degree of closeness.
I should like to apply a more rational process, one in which we can
admit not only arbitrary integral values of n but also arbitrary complex

values of w . Since we need to determine only one solution of the equation,

.
we shall seek, in particular, that value z = y w which lies within the

angle 2 n\n laid off on the axis of real numbers. Generalizating the ele-
mentary method mentioned above, we begin by dividing this angle into
v equal parts (v = 5 in Fig. 52), and by drawing circles intersecting the
dividing rays by circles which have the origin as common center and

1 See Ikosaeder, p. 55.

132 Algebra: Equations in the Field of Complex Quantities.

whose radii are measured by the numbers r = 1 , 2 , 3 , . . . In this way,
after choosing v, we find all the points

2i7t k /k = Q, 1, 2, . . ., v \
z = r-e" n v ( /=1 ,2,3,...

marked within the angular space, and we can at once mark in the
w plane the corresponding w values

w = z n = r n e v .

These will be the corners of a corresponding network (see Fig. 53)
covering the entire w plane and consisting of circles with radii l w , 2 n ,
3 n , . . . together with rays inclined to the real axis at angles of 0, 2 n\v ,

w-plane
^
z-plane

Fig. 53.

(^ 1) 2Ji[v. Let the given value of w lie either within or
on the contour of one of the meshes of this lattice, and suppose that W Q

n. -

is the lattice corner nearest to it. We know a value Z Q of ]/ze> is a corner
of the lattice in the z plane; hence the value we are seeking will be

We expand the right side by the binomial theorem, which we may con-
sider known, inasmuch as we are now, in reality, in the domain of
analysis

We can decide at once as to the convergence of this series if we look
upon it as the Taylor's development of the analytic function ^w and apply
the theorem that it converges within the circle which has W Q as

n .

center and which passes through the nearest singular point. Since }w
has only and oo as singular points, our development will converge if,
and only if, w lies within that circle about W Q which passes through the
origin, and we can always bring this about by starting, in the z plane,
with a similar lattice which may have smaller meshes, if necessary.
But in order that the convergence should be good, i.e., in order that the series

Uniformization of the Normal Irrationalities.

133

should be adapted to numerical calculation (w WQ)/W Q must be sufficiently
small. This can always be effected by a further reduction of the lattice.
This is really a very usable method for the actual calculation of numerical
roots.

Now is it worthy of remark that the numerical solution of the remaining
normal equations of the regular solids is not essentially more difficult, but
I shall omit the proof. If we apply, namely, the same method to our
normal equations, starting from the mapping upon the w sphere of two
neighboring triangles, there will appear, in place of the binomial series,
certain other series that are well known in analysis and are well adapted
to practical use, called the hypergeometric series. In the year 1877
I set up 1 this series numerically.

6. Uniformization of the Normal Irrationalities by Means of
Transcendental Functions

I shall now discuss another method of solving our normal equations
which is characterized by the systematic employment of transcendental
functions. Instead of proceeding, in each individual case, with series
developments in the neighborhod of a known solution, we try to re-
present, once for all, the whole set of number pairs (w , z) which satisfy
the equation, as single-valued analytic functions of an auxiliary variable:
or, as we say, to uniformize the irrationalities defined by the equation.
If we can succeed by using only functions which can easily be tabulated,
or of which one already has, perhaps, numerical tables, one can obtain
the numerical solution of the equation without farther calculation. I am
the more willing to discuss this connection with transcendental functions
because it sometimes plays a part in school instruction, where it still
often has a hazy, almost mysterious, aspect. The reason for this is that
one is still clinging to traditional imperfect conceptions, although the
modern theory of functions of a complex variable has provided perfect
clearness.

I shall apply these general suggestions first to the pure equation.
Even in the schools, we always use logarithms in calculating the positive
solution of z n = r , for real positive values of r. We write the equation
in the form z = e l sr/n , where logr stands for the positive principal
value. The logarithmic tables supply first log r, and then, conversely, z
is the number that corresponds to log r/n. Moreover, we ordinarily use
10 as base instead of e. This solution can be extended immediately to
complex values. We satisfy the equation

z n = w ,

[ l Weiteve Untersuchungen uber das Ikosaeder, Mathematische Annalen, vol. 12,
p. 515. See also Klein, F., Gesammelte Mathematische Abhandlungen, vol. 2, p. 331
et seq.]

Algebra: Equations in the Field of Complex Quantities.

by putting x equal to the general complex logarithm, log w , after which
we obtain w and z actually as single-valued analytic functions of x :

w = e x , z = e n

In view of the many-valuedness of x = log w, which we shall study
later in detail, one obtains here for the same w precisely n values of z .
We call x the uniformizing variable.

Since the tables contain only the real logarithms of real numbers,
we are apparently unable to read off immediately the value of the given
solution. But by the aid of a simple property of logarithms, we can
reduce the calculation to the use of trigonometric tables which are accessible
to everybody. If we put

w = u -f- iv =

then the first factor, as a positive real number, has a real logarithm,
the second, as a number of absolute value 1 , a pure imaginary logarithm
i<p (i.e., the second factor is equal to e iv ), and we obtain <p from
the equation

u v

(a)

This gives x = log w = log | }u* + v 2 \ +iq>, and the root of the equation
is therefore

-- loglV^+^l Liy
z = e n = e n *e n

i. e., we have

X1 v n log I Vu* + v* I / qj , . . m\

(b) z = yu + iv = e n (cos + tsm--j.

Since from (a) und then z from (b).
We have obtained this "trigonometric solution" from the logarithms of
complex numbers in an entirely natural way. However, if we assume
that these are not known and try to develop this trigonometric solution,
as is done in the schools, it must appear as something entirely foreign
and unintelligible.

Occasionally it becomes necessary to find roots of numbers that are
not real. Thus, in school instruction, such roots must be found in the
so called Cardan's solution of the cubic equation about which I should
like to interpolate here a few remarks. If this equation is given in the
reduced form
(1) x* + px q = 0,

Uniformization of the Normal Irrationalities.

then the formula of Cardan states that its three roots x lt x 2 , # 3 are
contained in the expression

(2)

Since every cube root is three valued, this expression has, all told,
nine values, in general all different; among these, x lt x%, x$ are deter-
mined by the condition that the product of the two cube roots employed
each time is p/^ . If we replace the coefficients p , q in the well known
manner by their expressions as symmetric functions of x l9 x z , x 3 , and
if we note that the coefficient of x 2 vanishes, that is, x l + x% + x$ = 0,
we get

q I p (#1 #2) (^2 ^3) v^s ^i)

T 27 ~~ 7 108 '

that is, the radicand of the square root is, to within a negative factor,
the discriminant of the equation. This shows at once that it is negative
when all three roots are real, but positive when one root is real and the other
two conjugate imaginary. It is precisely in the apparently simplest case
of the cubic equation, namely when all the roots are real, that the
formula of Cardan requires the extraction of the square root of a nega-
tive number, and hence of the cube root of an imaginary number.

This passage through the complex must have seemed something
quite impossible to the mediaeval algebraists at a time when one was
still far removed from a theory of complex numbers, 250 years before
Gauss gave his interpretation of them in the plane! One talked of the
"Casus irreducibilis" of the cubic equation and said that the Cardan
formula failed here to give a reasonable usable solution. When it was
discovered later that it was possible, precisely in this case, to establish
a simple relation between the cubic equation and the trisection of an
angle, and to get in this way a real "trigonometric solution* ' in place
of the defective Cardan formula, it was believed that something new
Had been discovered which had no connection with the old formula.
Unfortunately this is the position taken occasionally even today in
elementary instruction.

In opposition to this view, I should like to insist here emphatically
that this trigonometric solution is nothing else than the application, in
calculating the roots of complex radicands, of the process which we have
just discussed. It is obtained therefore in a perfectly natural way in
this case, where the cube root has a complex radicand, if we transform
the Cardan formula, for numerical calculation, in the same convenient
way that one pursues in school for the case of the real radicand. In fact,
let us suppose

136

Algebra: Equations in the Field of Complex Quantities.

where p must be negative if q is real. If we then write the first cube
root in (2) in the form

-+.-I !/--!

We note that its absolute value value (as positive cube root of the
value V p*/27 of the radicand) is equal to | V pft ; but since the
product of this by the second cube root is equal to p/3 , that second
cube root must be the conjugate complex of this, and the sum of the
two, i.e., the solution of the cubic equation, is simply twice the real
part, that is,

Now let us apply the general procedure of p. 134. We write the
radicand of the cube root, after separating out its absolute value, in
the form

I/ ?2

t> B

p*-

IF 4 27

]/ ^

I *

I/ ^ 3

V 27

r 27

and determine

an angle

9? from the equations

ro^/Y)

r 4 27

TJJ" '

L 1

I'"

y-i

Then, since the positive cube root of |V p*/27\ is |V
root takes the form

, our cube

and hence, remembering that (p is determinate only to within multiples
of 2^, we obtain

' COS J

But this is the usual form of the trigonometric solution.

I should like to take this opportunity to make a remark about the
expression "casus irreducibilis" . "Irreducible" is used here in a sense
entirely different from the one in use today and which we shall often
use in these lectures. In the sense here used it implies that the solution
of the cubic equation cannot be reduced to the cube roots of real numbers.
This is not in the least the modern meaning of the word. You see how
the unfortunate use of words, together with the general fear of cBmplex
numbers, has created at least the possibility for a good deal of misunder-

Uniformization of Normal Irrationalities.

137

standing in just this field. I hope that my words may serve as a preven-
tive, at least among you.

Let us now inquire briefly about uniformization by means of trans-
cendental functions in the case of the remaining normal irrationalities.
In the dihedral equation

z n + = 2w

we put simply

w = cos <f .

De Moivre's formula shows that the equation is then satisfied by

w . . . w

z = cos + & sin .

n n

Since all values of 9? + 2 k n and of 2 k n q> gi ve the same value of w
this formula gives, in fact, for every w, 2n values of z, which we can
write

(p + 2kn . . . cp 4- 2kn ..

z = cos 5^ t sin^ ------ . (k = 0, 1 , 2, . . ., n - 1)

In the case of the equations of the octahedron, tetrahedron, and
icosahedron these "elementary" transcendental functions do not suffice.
However, we can obtain the corresponding solution by means of elliptic
modular functions. Although one may not consider this solution as
belonging to elementary methematics, I should, nevertheless, like to
give, at least, the formulas 1 which relate to the icosahedron. They are,
namely, closely related to the solution of the general equation of degree
five by means of elliptic functions, to which allusion is always made
in textbooks and about which I shall have something to say later by
way of explanation. The icosahedral equation had the form (see pp. 130,
126)

Now we identify w with the absolute invariant / from the theory of
elliptic functions and think of / as a function of the period quotient
w = oVojg (in Jacobi's notation i K'jK), i.e., we set

t .

zf(e lf o> 2 )

where g 2 and A are certain transcendental forms of dimension 4 and
12, respectively, in o^ and co 2 , which play an important role. If we
introduce the usual abbreviation of Jacobi

q = e iato = e TK

t 1 See Mathematische Annalen, vol. 14 (1878/79), p. Ill et seq., or Klein,
Gesammelte Abhandlungen, vol. 3, p. 13 et seq., also Ikosaeder, p. 131.]

Algebra: Equations in the Field of Complex Quantities.

the roots z of the icosahedral equation will be given by the following
quotients of ft functions

If we take into account that co as a function of w, coming from the
first equation, is infinitely many- valued, then this formula yields in
fact all sixty roots of the icosahedral equation for a given w.

7. Solution in Terms of Radicals

There is one question in the theory of the normal equations which
I have not yet touched, namely, whether or not our normal equations
yield algebraically anything that is essentially new; and whether or
not they can be resolved into one another or, in particular, into a sequence
of pure equations. In other words, is it possible to build up the solution*
of these equations in terms of w by means of a finite number of radical
signs, one above another?

So far as the equations of the dihedron, tetrahedron, and octahedron
are concerned, it is easy to show, by means of algebraic theory, that
they can be reduced, in fact, to pure equations. It will be sufficient
if I give the details here for the dihedral equation only:

z n + ^ = 2w.

If we set:

* W = C,
the equation goes over into

t 2 - 2w+ \ =0.
It follows from this that

= w /w 2 1 ,
and consequently

-which is the desired solution by means of radicals.

On the other hand, however, the icosahedral equation does not admit
such a solution by means of radicals, so that this equation defines an
essentially new algebraic function. I am going to give you a particularly
graphic proof of this, which I have recently published (Mathematische
Annalen, Vol. 61 [1905]), and which follows from consideration of the
familiar function theoretic construction of the icosahedral function z (w) .
For this purpose I shall need the following theorem, due to Abel, a
proof of which you will find in every treatise on algebra: // the solution
of an algebraic equation can be expressed as a sequence of radicals, then
every radical of the sequence can be expressed as a rational function of
the n roots of the given equation.

Solution in Terms of Radicals.

Let us now apply this theorem to the icosahedral equation. If we
assume its root z can be expressed as a sequence of roots of rational
functions of the coefficients, i.e., of rational functions of w t then every
radical in the sequence is a rational function of the sixty roots:

R (z l , z 2 , . . . , 2 60 ) .

(We shall show that this leads to a contradiction.) In the first place,
we can replace this expression by a rational function R (z) of z alone
since all the roots can be derived from any one of them by a linear
substitution. Let us now convert this R (z) into a function of w by
writing for z the sixty-valued icosahedral function z (w) , and consider
the result. Since every circuit in the w plane which returns z to its
initial value must of necessity return R (z) also to its initial value, it
follows that +R[z(w)] can have branch points only at the positions
w = 0, 1 , oo (where z (w) has branch points), and the number of leaves
of the Riemann surface for R [z (w)] which are cyclically connected at
each of these positions must be a divisor of the corresponding number
belonging to z (w). We know that this number is 3, 2, 5 at the three
positions, respectively. Hence every rational function R (z) of an icosa-
hedral root, and consequently every radical which appears in the assumed
solution, considered as function of w, can have branch points, if at all,
only a,tw = Q,w = \,w=oo. If branching occurs, then there must
be three leaves connected at w = 0, two at w = \ , and five at w = oo,
since 3,2,5 have no divisor other than 1 .

We shall now see that this result leads to a contradiction. To this
end let us examine the innermost radical which appears in our hypothe-
tically assumed expression for z (w). Its radicand must be a rational
function P (w) . We can assume that the index of the radical is a prime
number p, since we could otherwise build it up out of radicals with
prime indices. Moreover P (w) cannot be the ^?-th power of a rational
function Q (w) of w, for if it were, our radical would be superfluous,
and we could direct our attention to the next really essential radical.

Let us now see what kind of branchings the function y P (w) can
have. For this purpose it will be convenient to write it in the homo-
geneous form

where g and h are forms of the same dimension in the variables w l , w%
(w = Wi/w 2 ) . According to the fundamental theorem of algebra we
can separate g and h, into linear factors and write

where

140 Algebra: Equations in the Field of Complex Quantities.

since the numerator and the denominator are of the same degree. Not
all the exponents a, ft, ...,<*', $' . . . can be divisible by p, since P
would then be a perfect p-th power. On the other hand, a + /? + ...
' /?' . . . is equal to zero, and is therefore divisible by p.
Consequently at least two of these numbers are not divisible by p.
It follows that the zeros of both the corresponding linear factors must

be branch points of /P(z0), at each of which p leaves are cyclically
connected. But herein lies the contradiction of the previous theorem,

which, of course, must be equally valid for V P (w) . For we enumerated
at that time all possible branch points, and we found among them no
two at which the same number of leaves were connected. Our assumption
is therefore not tenable, and the icosahedral equation cannot be solved
by radicals.

This proof depends essentially upon the fact that the numbers 3 , 2 , 5
which are characteristic for the icosahedron have no common divisor.
When such a common divisor appears, as in the case of the numbers 3 ,
2, 4 of the octahedron, it is at once possible to have rational functions
R [z (w)] which exhibit the same kind of branching at two points, e.g.,
one in which two leaves are connected at 1 and at oo, and these can
then be really represented as roots of a rational function P (w) . It is
in this way that the solution by means of radicals comes about in the
case of the octahedron and tetrahedron (with the numbers 3*2,3),
and of the dihedron (2,2,w).

I should like to show you here how slightly the language used in
wide mathematical circles keeps pace with knowledge. The word "root"
is used today nearly everywhere in two senses: once for the solution
of any algebraic equation, and, secondly, in particular, for the solution
of a pure equation. The latter use, of course, dates from a time when
only pure equations were studied. Today it is, if not actually harmful,
at least rather inconvenient. Thus it seems almost a contradiction to
say that the "roots' ' of an equation cannot be expressed by means of
radical signs. But there is another form of expression which has lingered
on from the beginnings of algebra and which is a more serious source of
misunderstanding, namely, that algebraic equations are said to be "not
algebraically solvable", if they cannot be solved in terms of radicals
i. e. if they cannot be reduced to pure equations. This use is in immediate
contradiction with the modern meaning of the word "algebraic". Today
we say that an equation can be solved algebraically when we can reduce
it to a chain of simplest algebraic equations in which one controls the
dependence of the solutions upon the parameters, the relation of the
different roots to one another, etc. as completely as one does in the
case of the pure equation. It is not at all necessary that these equations
should be pure equations. In this sense we may say that the icosahedral

Reduction of General Equations to Normal Equations. \^\

equation can be solved algebraically, for our discussion shows that we
can construct its theory in a manner that meets all the demands men-
tioned above. The fact that this equation cannot be solved by radicals
lends it special interest by suggesting it as an appropriate normal
equation to which one might try to reduce, (i. e., completely solve)
still other equations which are in the old sense algebraically unsolvable.
The last remark leads us to the last section of this chapter, in which
we shall try to get a general view of such reductions.

8. Reduction of General Equations to Normal Equations

It turns out, namely, that the following reductions are possible:
The general equation of the third degree to the dihedral equation forn = ^\
The general equation of the fourth degree to the tetrahedral or to the
octahedral equation]

The general equation of the fifth degree to the icosahedral equation.
This result is the most recent triumph of the theory of the regular
bodies which have always played such an important r61e since the
beginning of mathematical history, and which have a decisive influence
in the most widely separated fields of modern mathematics.

In order to show you the meaning of my general assertion I shall
go somewhat more into details for the equation of degree three, without,
however, fully proving the formulas. We again take the cubic equation
in the reduced form

(1) # 3 + px q = 0.

Denoting solutions by x lt x 2 , x$, we try to set up a rational function z
of them which undergoes the six linear substitutions of the dihedron for
n = 3 when we interchange the Xi in all six possible ways. The values
that z should take on are

z, ez, e z z, , ," I where = <
z z z \

It is easily seen that

(2} z Xl + sx * + '

\ ' V _L r-2^ l_

satisfies these conditions. The dihedral function z* + \/z 3 of this quantity
must remain unaltered by all the interchanges of the Xk, since the
six linear substitutions of the z leave it unchanged. Hence, by a well
known theorem of algebra, it must be a rational function of the co-
efficients of (1). A calculation shows that

(3) * + . = -27^-2.

Conversely, if we solve this dihedral equation, and if z is one of its
roots, we can express the three values x l9 x%, x 3 rationally in terms of

142 Algebra: Equations in the Field of Complex Quantities.

z, p, and q by means of (2) and the well known relations

Doing this, we find

X *=P

v __3?

Thus, as soon as the dihedral equation (3) has been solved, the formulas
(4) give at once the solution of the cubic (1).

In the same way we may reduce the general equations of the fourth
and fifth degrees. The equations would be, of course, somewhat longer,
but not more difficult in principle. The only new thing would be that
the parameter w of the normal equation, which was expressed above

rationally in the coefficients of the equation \2w= 27^2],

would now contain square roots. You will find this theory for the
equation of degree five given fully in the second part of my lectures
on the icosahedron. Not only are the formulas calculated, but also the
essential reasons for the appearance of the equations are explained.
Finally, let me say a word about the relation of this development
to the usual presentation of the theory of equations of the third, fourth,
and fifth degree. In the first place, we can obtain the usual solutions
of the cubic and biquadratic from our formulas by appropriate reduc-
tions, if we use the solutions of the equations of the dihedron, octahedron,
and tetrahedron in terms of radicals. In the case of equations of degree
five, most of the textbooks confine themselves unfortunately to the
establishment of the negative result that the equation cannot be solved
by radicals, to which is then added the vague hint that the solution
is possible by elliptic functions, to be exact one should say elliptic
modular functions. I take exception to this procedure because it ex-
hibits a one-sided contrast and hinders rather than promotes a real
understanding of the situation. In view of the preceding survey, using
first algebraic and then analytic language, we may say:

1 . The general equation of the fifth degree cannot be reduced, indeed,
to pure equations, but it is possible to reduce it to the icosahedral equation
as the simplest normal equation. This is the real problem of its algebraic
solution.

2. The icosahedral equation, on the other hand, can be solved by elliptic
modular functions. For purposes of numerical calculation, this is the
full analog of the solution of pure equations by means of logarithms.

Reduction of General Equations to Normal Equations.

This supplies the complete solution of the problem of the equation
of fifth degree. Remember that when the usual road does not lead to
success, one should not be content with this determination of impossi-
bility, but should bestir oneself to find a new and more promising route.
Mathematical thought, as such, has no end. If someone says, to you that
mathematical reasoning cannot be carried beyond a certain point, you
may be sure that the really interesting problem begins precisely there.

In conclusion, it might be remarked that these theories do not stop
with equations of degree five. On the contrary, one can set up analogous
developments for equations of the sixth and higher degrees if one will
only make use of the higher-dimensional analogs of the regular bodies.
If you are interested in this, you might read my article 1 Ober die Auf-
losung der allgemeinen Gleichung funften und sechsten Grades*. In con-
nection with this article the problem was successfully attacked by
P. Gordan 2 and A. B. Coble 3 . The investigation is somewhat simplified
in the latter memoir 4 .

1 Journal fiir Mathematik,.vol. 129 (1905), p. 151 ; and Mathematische Annalen,
vol. 61 (1905), P- 50.

* Concerning the solution of the general equation of fifth and of sixth degree.

2 Mathematische Annalen, vol. 61 (1905), p. 50; and vol. 68 (1910), p. 1.

3 Mathematische Annalen, vol. 70 (1911), p- 337-

4 See also Klein, F., Gesammelte Mathematische Abhandlungen, vol. 2,
p. 502-503.

Part Three

Analysis

During this second half of the semester we shall select certain chapters
in analysis which are important from our standpoint and we shall
discuss them as we did arithmetic and algebra. The most important
thing for us to discuss will be the elementary transcendental functions,
i. e. logarithmic and exponential functions and trigonometric functions,
since they play an important part in school instruction. Let us begin
with the first.

I. Logarithmic and Exponential Functions

Let me recall briefly the familiar curriculum of the school, and the
continuation of it to the point at which the so called algebraic analysis
begins.

1. Systematic Account of Algebraic Analysis

One starts with powers of the form a = b c , where the exponent c
is a positive integer, and extends the notion step by step for negative
integral values of c, then for fractional values of c, and finally, if cir-
cumstances warrant it, to irrational values of c. In this process the
concept of root appears as that of a particular power. Without going
into the details of involution, I will only recall the rule for multiplication

which reduces the multiplication of two numbers to the addition of
exponents. The possibility of this reduction, which, as you know, is
fundamental for logarithmic calculation, lies in the fact that the fun-
damental laws for multiplication and addition are so largely identical,
that both operations, namely, are commutative as well associative.
The operation inverse to that of raising to a power yields the
logarithm. The quantity c is called the logarithm of a to the base 6:

c
(W

At this point a number of essential difficulties appear which are
usually passed over without any attempt at explanation. For this reason

Systematic Account of Algebraic Analysis.

I shall try to be especially clear at this point. For the sake of convenience
we shall write x and y instead of a and c , inasmuch as we wish to study
the mutual dependence of these two numbers. Our fundamental equa-
tions then become

x = b y , y = log*.

(b)

Let us first of all notice that b is always assumed to be positive. If b
were negative, x would be alternately positive and negative for integral
values of y, and would even include imaginary values for fractional
values of y , so that the totality of number pairs (x , y) would not give
a continuous curve. But even with b > one cannot get along without
making stipulations that appear to be quite arbitrary. For if y is
rational, say y=m/n f where m and n are integers prime to each other,

x = b mfn is, as you know, defined to be y b m and it has accordingly n
values, of which, for even values of n, we should have two to deal with
even if we confined ourselves to real numbers. It is customary to
stipulate that x shall always be the positive root, the so-called principal
root.

If you will permit me to use, somewhat prematurely, the familiar
graph of the logarithm y = logx (Fig. 54), you will see that neither
the above stipulation nor its suit-
ableness is by any means self-evident. "* V * S 4^ >
If y traverses the dense set of rational
values, the corresponding points whose
abscissas are the positive principal
values x = b y constitute a dense set
on our curve. If, now, when the de-
nominator n of y is even, we should Fig. 54.
mark the points which correspond to

negative values of x, we have a set of points which would be, one might
say, only half so dense, but nevertheless dense on the curve which is the
reflection in the y axis of our curve [y = log (x)]. If we now admit
all real, including irrational, values of y , it is certainly not immediately
clear why the principal values which we have been marking on the right
now constitute a continuous curve and whether or not the set of negative
values which we have marked on the left is similarly raised to a con-
tinuum. We shall see later that this can be made clear only with the
profounder resources of function theory, an aid which is not at the com-
mand of the elementary student. For this reason, one does not attempt
in the schools to give a complete exposition. One adopts rather an
authoritative convention, which is quite convincing to the pupils,
namely that one must take b > and must select the positive principal
values of x, that everything else is prohibited. Then the theorem follows,

Klein, Elementary Mathematics. 10

146 Analysis: Logarithmic and Exponential Functions.

of course, that the logarithm is a single- valued function defined only for
a positive argument.

Once the theory is carried to this point, the logarithmic tables are
put into the hands of the pupil and he must learn to use them in practical
calculation. There may still be some schools in my school days this
was the rule where little or nothing is said as to how these tables
are made. That was despicable utilitarianism which is scornful of every
higher principle of instruction, and which we must surely and severly
condemn. Today, however, the calculation of logarithms is probably
discussed in the majority of cases, and in many schools indeed the
theory of natural logarithms and the development into series is taught
for this purpose.

As for the first of these, the base of the system of natural logarithms
is, as you know, the number

* = lim(l + } H = 2.7182818

n=<x) \ n i

This definition of e is usually, in imitation of the French models, placed
at the very beginning in the great text books of analysis, and entirely
unmotivated, whereby the really valuable element is missed, the one
which mediates the understanding, namely, an explanation why pre-
cisely this remarkable limit is used as base and why the resulting
logarithms are called natural. Likewise the development into series is
often introduced with equal abruptness. There is a formal assumption
of the development

log (1 + *) = a + a v % + a 2 * 2 H ,

the coefficients a , a l9 . . ., are calculated by means of the known pro-
perties of logarithms, and perhaps the convergence is shown for | # | < 1 >
But again there is no explanation as to why one would ever even suspect
the possibility of a series development in the case of a function of such
arbitrary composition as is the logarithm according to the school de-
finition.

2. The Historical Development of the Theory

If we wish to find all the fundamental connections whose absence
we have noted, and to ascertain the deeper reasons why those apparently
arbitrary conventions must lead to a reasonable result, in short, if we
wish really to press forward to a full understanding of the theory of
logarithms, it will be best to follow the historical development in its
broad outlines. You will see that it by no means corresponds' to the
practice mentioned above, but rather that this practice is, so to speak,
a projection of that development from a most unfavorable standpoint.

We shall mention first a German mathematician of the sixteenth
century, the Swabian, Michael Stifel, whose Arithmetica Integra appeared

The Historical Development of the Theory.

in Niirnberg in 1544. This was the time of the first beginnings of our
present algebra, a year before the appearance, also in Niirnberg, of the
book by Cardanus, which we have mentioned. I can show you this
book, as well as most of those which I shall mention later, thanks to
our unusually complete university library. You will find that it uses,
for the first time, operations with powers where the exponents are any
rational numbers, and, in particular, emphasizes the rule for multi-
plication. Indeed, Stifel gives, in a sense, the very first logarithmic
table (see p. 250) which, to be sure, is quite rudimentary. It contains
only the integers from 3 to 6 as exponents of 2 , along with the corres-
ponding powers |- to 64. Stifel appears to have appreciated the signi-
ficance of the development of which we have here the beginning. He
declares, namely, that one might devote an entire book to these re-
markable number relations.

But in order to make logarithms really available for practical calcula-
tion Stifel lacked still an important device, namely, decimal fractions ;
and it was only when these became common property, after 1600, that
the possibility arose of constructing real logarithmic tables. The first
tables were due to the Scotchman Napier (or Neper), who lived 1550 1 61 7.
They appeared in 1614, in Edinburgh, under the title Mirifici logarith-
morum canonis descriptio, and the enthusiasm which they aroused is
evidenced by the verses with which different authors in their prefaces
sang the virtues of logarithms. However, Napier's method for calculating
logarithms was not published until 1619, after his death, as Mirifici
logarithmorum canonis construct 1 .

The Swiss, Jobst Biirgi (15521632), had calculated a table in-
dependently of Napier, which did not appear, however, until 1620, in
Prag, under the title Arithmetische und geometrische Progresstabuln. We,
in Gottingen, should have a peculiar interest in Biirgi, as one of our
countrymen, since he lived for a long time in Cassel. In general, Cassel,
particularly the old observatory there, has been of importance for the
development of arithmetic, astronomy, and of optics prior to the
discovery of infinitesimal calculus, just as Hannover became important
later as the home of Leibniz. Thus our immediate neighborhood was
historically significant for our science long before this university was
founded.

It is very instructive to follow the train of thought of Napier and
Biirgi. Both start from values of x = b y for integral values of y and
seek an arrangement whereby the numbers % shall be as close together
as possible. Their object was to find for every number #, as nearly as
possible, a logarithm y . This is achieved today, in school, by considering
fractional values of y, as we saw before. But Napier and Biirgi, with the

1 Lugduni 1620. There is a later edition in phototype. (Paris 1895-)

10*

148 Analysis: Logarithmic and Exponential Functions.

intuition of genius, avoided the difficulties which thus present themselves
by grasping the thing by the smooth handle. They had, namely, the
simple and happy thought of choosing the base b close to one, when,
in fact, the successive integral powers of b are close to one another.
Biirgi takes

b = 1.0001,
while Napier selects a value less than one, but still closer to it:

6 = 1- 0.0000001 = 0.9999999-

The reason for this departure by Napier from the method of today is
that he had in mind the application to trigonometric calculation, where
one has to do primarily with logarithms of proper fractions (sine and
cosine) and these are negative for b > 1 but positive for b < 1 . But
with both investigators the chief thing was that they made use only
of integral powers of this b and so avoided, completely, the many valued-
ness which embarrassed us above.

Let us now calculate, in the system of Biirgi, the powers for two
neighboring exponents, y and y + 1 :

x + Ax =

By subtraction, then, we have

A x = (1.0001)" (1.0001 1) = ^

or, writing Ay for the differences, 1, of the values of the exponent:

(la) 7* = -

v ' Ax x

We have thus obtained a difference equation for the Biirgi logarithms,
one which Biirgi himself used directly in the calculation of his tables.
After he had determined the oc corresponding to a y he obtained the
following % belonging to y + 1 by the addition of #/10 4 . In the same
way it follows that the logarithms of Napier satisfy the difference
equation

db) = -

v ' Ax x

In order to see the close relationship between the two systems, we
need only write for y on the one hand y/10 4 , on the other hand y/10 7 , i.e.,
we need only displace the decimal point in the logarithm. If we denote
the new numbers so obtained simply by y, we shall have in each case
a series of numbers which satisfy the difference equation

(2)

v '

-
Ax

and in which the values of y proceed by steps of 0.0001 in the one case
and of 0.0000001 in the other.

The Historical Development of the Theory.

149

If, for the sake of convenience, we now make use of the graph of
the continuous exponential curve (we ought really to obtain it as the
result of our discussion) we shall have a tangible representation of the
points which correspond to the number series of Napier and of Biirgi.
These points will be the corners of a stairway inscribed in one of the
two exponential curves

(3) x = (1.0001) 10000 *, and x = (0.9999999) 10000000y ,

respectively, where the risers have the constant value Ay =0.0001 and
Ay 0.0000001 in the two systems, respectively (see Fig. 55).

We can get another geometric interpretation in which we do not
need to presuppose the exponential curve, which will rather point out
the natural way to obtain that curve, if we
replace the difference equation (2) by a summa-
tion equation, that is, if we integrate it, in a sense :

(4) ,-V"

Fig. 55-

During this summation increases disconti-
nuously, from unity on, by such steps that the
corresponding A r\ = J/ is always constant and

equal to 10 ~ 4 and 10 ~ 7 respectively, so that A = f/10 4 and /10 7 , in
the two cases. With the last step f attains the value x. Once can easily
give geometric expression to this procedure. For this purpose let us
draw the hyperbola rj = 1/f in an
f ?7 plane (see Fig. 56) and, begin-
ning at ^ = 1 , construct succes-
sively on the | axis the points that
are given by the law of progression
A | = /10 4 (confining ourselves to
the Biirgi formulation). The rect-
angle of altitude 1/f erected upon
each of the intervals so ob-
tained will have the constant area
/! i/| = i/io 4 . The Biirgi logarithm will then be, according to (4),
the 10 4 -fold sum of all these rectangles inscribed in the hyperbola and
lying between 1 and x. A similar result is obtained for the logarithm
of Napier.

Proceeding from this last representation, one is led immediately to
the natural logarithm if, instead of the sum of the rectangles, one takes
the area under the hyperbola itself between the ordinates f = 1 and
. = x (shaded in the figure). This finds expression in the well-known
formula

150 Analysis: Logarithmic and Exponential Functions.

This was, in fact, the historical way, and the decisive step was taken
about 1650, when analytic geometry had become the common possession
of mathematicians and the infinitesimal calculus was achieving the
quadrature of known curves.

If we desire to use this definition of the natural logarithm as our
starting point, we must, of course, convince ourselves that it possesses
the fundamental property of replacing the multiplication of numbers
by the addition of logarithms ; or, in modern terms, we must show that
the function

defined thus by means of the area under the hyperbola, has the simple
addition theorem

/(*i) +/(* 2 ) = /(*i'*2)

In fact, if we vary x l and x 2> then, according to the definition of an
integral, the increments of the two sides dx^x^ + dx 2 /x 2 and
d (x l x 2 )/(x l # 2 ) are equal. Consequently / (x^ + / (x 2 ) and / (x x z )
can differ only by a constant, and this turns out to be zero when
we put x l = 1 (since / (1) = 0).

If we wish to determine the "base" of the logarithms obtained in
this way, we need only notice that the transition from the series of
rectangles to the area under the hyperbola can be made by changing
the increment A = f /10 4 to A = /n and allowing n to become infinite.
This is the same thing as replacing the Biirgi sequence x (1.0001) 10000z/
by x = (1 + \\n) ny , where ny becomes infinite through integral values.
According to the general definition of a power, this amounts to saying
that x is the y-th power of (1 + \jn) n . Accordingly it seems plausible
to say that the base is lim (1 + \/n) n , the very limit which is ordinarily

W=oo

assumed at the start as the definition of e. It is interesting to note,
moreover, that Biirgi's base (1.0001) 10000 = 2.718146 coincides with e
to three decimal places.

Let us now examine the historical development of the theory of
the logarithm after Napier and Biirgi. First of all I shall make the
following statements.

1. Mercator, whom we have already met in these pages (see p. 81)
was one of the first to make use of the definition of the logarithm by
means of the area of the hyperbola. In his book Logarithmotechnica
of 1668, as well as in articles in the Philosophical Transactions of the
London Royal Society in 1667 and 1668, he shows, by means of the
same argument which I have just given you in modern terms, that

//
dSIS differs from the common logarithm with the base 10, which

was already the base used in calculations, only by a constant factor,

The Historical Development of the Theory.

the so called modulus of the system of logarithms. Moreover he had
already introduced 1 the name "natural logarithm" or "hyperbolic
logarithm". But the greatest achievement of Mercator was the setting
up of the power series for the logarithm, which he obtained (essentially,
at least) from the integral representation by dividing out and integrating
term by term. I mentioned this to you (p. 81) as an epochmaking
advance in mathematics.

2. In that same connection, I told you also that Newton had taken
up these ideas of Mercator 1 s and had enriched them with two important
results, namely, the general binomial theorem and the method for the
reversion of series. This last appeared in a work of Newton's youth
De analysi per aequationes numero terminorum infinitas which appeared
late in print but which from 1669 on was distributed in manuscript
form 2 . In this 3 Newton derives the exponential series

for the first time by reverting Mercator' s series for y = log A;. This
yields, as the number whose natural logarithm y = \

_ l 1 1

6 " 1 + 1 ! + 27 + 3"! + ' " '

and it is now easy, with the aid of the functional equation for the
logarithm, to show that, for every real rational y, # is one of the values
of e y , and in fact the positive value, in the sense of the customary
definition of power. We shall go into this more in detail later on. The
function y = log % thus turns out to be precisely what one would
call the logarithm of x to the base e , according to the ordinary definition,
in which e is defined by means of the series and not as lim (1 + \/n) n .

n=oo

3. Brook Taylor could follow a more convenient path in deriving
the exponential series, after he had devised the general series-development
which bears his name, which appeared in his work Methodus Incremen-
torum* and of which we shall have much to say later on. He could
then use the relation

d\ogx _ J_
dx ~~ ~x '

which is implied in the integral definition of the logarithm, infer from
it the inverse relation

de * y

-= = e y
dy

1 Philosophical Transactions of the Royal Society of London, vol. 3 (1668),
p. 761.

2 Newton, I.: Opuscula, Tome I, op. 1, Lausanne 1744. Appeared first in 1711.

3 Loc. cit., p. 20.

4 London, 1715.

152 Analysis: Logarithmic and Exponential Functions.

and so write down at once the exponential series as a special case of
his general series.

We have already seen (p. 82) how this productive period was followed
by the period of criticism, I should almost like to say the period of
moral despair, in which every effort was directed toward placing the
new results upon a sound basis and in separating out what was false.
Let us now see what attitude was taken toward the exponential function
and the logarithm in the books of Euler and Lagrange, which tended
in this new direction.

We shall begin with Euler 's Introductio in analysin infinitorum 1 .
Let me, first of all, praise the extraordinary and admirable analytic
skill which Euler shows in all his developments, noting, however, at
the same time, that he shows no trace of the rigor which is demanded
today.

At the head of his developments Euler places the binomial theorem

in which the exponent I is assumed to be an integer. Now integral
exponents are not considered in the Introductio. This development is
specialized for the expression

/. . \ \ n V

in which ny is integral. He then allows n to become infinite, applies
this limit process to each term of the series, thinks of e as defined by
lim (1 + l/w) n , and so obtains the exponential series

To be sure, Euler is not in the least concerned here as to whether or not the
individual steps in this process are rigorous, in the modern sense; in
particular, whether the sum of the limits of the separate terms of the
series is really the limit of the sum of the terms, or not. Nowthis derivation
of the exponential has been, as you know, a model for numerous text-
books on infinitesimal calculus, although, as time went on, the different
steps have been more and more elaborated and their legitimacy put to
the test of rigor. You will see how influential Euler' s work has been
for the entire course of these things if you recall that the use of the
letter e for that important number is due to him. "Ponamus autem
brevitatis gratia pro numero hoc 2.71828 . . . constanter litteram e" ,
as he writes on page 90.

1 Lausanne, 1748, Caput VII, p. 85 et seq. Translation by Maser, Berlin 1885,
p. 70. [See also vol. VIII (1923) of Euler's Works, edited by F. Rudio, A. Krazer,
and P. Stackel.]

The Historical Development of the Theory. >jci

I might add that Euler immediately follows this with an entirely
analogous derivation of the series for the sine and cosine. For this pur-
pose he starts with the development of sin (p in powers of sin (fpfn)
and lets n become infinite. This is nothing else than a limit process
applied to the binomial theorem, as is evident if one obtains the power
series in question from De Moivre's formula:

, / <P , - <p\ n i <p\ n ( w\ n

cos<p + % sm<p = Jcos-^ + % sm-J-j - ^cos J (1 + t tg-Jj .

Let us now consider Lagrange' s Theorie des fonctions analytiques 1 .
Again it is to be noted that questions of convergence are treated, at
most, only incidentally. I have already stated (p 83) that Lagrange
considers only those functions that are given by power series, and defines
their differential quotients formally by means of the derived power series.
Consequently the Taylor's series

is for him simply the result of a formal reordering of the series for
/ (x + h) proceeding originally according to powers of x + h. Of course,
if one wishes then to apply this series to a given function, one ought
really to show in advance that this function is analytic, i.e., that it can
be developed into a power series.

Lagrange begins with the investigation of the function / (x) = x n ,
for rational n , and determines /' (x) as the coefficient of h in the expansion
of (x + h) n , the first two terms of which he thinks of as calculated.
Then, by the same law, he obtains at once /" (x) , /'" (x) , . . . , and the
binomial expansion of (x + h) n appears as a special case of Taylor's
series for / (x + h) . Moreover, let me note expressly that Lagrange does
not give special consideration to the case of irrational exponents, but
rather looks upon it as obviously settled when he has considered all
rational values. It is interesting to contemplate this fact, since it is
upon the rigorous justification of precisely this sort of transition that
the greatest importance is laid today.

Lagrange uses these results in a similar treatment of the function
/ (x) = (1 + 6)*. By recording the binomial series for (1 + b) x+h he
finds, namely, f (x) as the coefficient of h, then determines /"(#),
/'" (x) , . . . according to the same law, and forms, finally, the Taylor
series for / (x + h) = (1 + b} x+h . He is then in possession, for h = 0,
of the desired exponential series.

I should like now to finish this brief historical sketch, in which
I have, of course, mentioned only names of the very first rank, by in-
dicating what essentially new turns came with the nineteenth century.

1 Paris, 1797, Reprinted in Lagrange, CEuvres, vol. 4. Paris 1881. Compare
especially chapter 3, p. 34 et seq.

154 Analysis: Logarithmic and Exponential Functions,

1 . At the head of this list I should place the precise ideas concerning
the convergence of infinite series and other infinite processes. Gauss
takes precedence here with his Abhandlung uber die hypergeometrische
Reihe* in 1812 (Disquisitiones generates circa seriem infinitam
1 + [(a b)/(i c)] x + - ) l . After him comes Abel with his memoir on the
binomial series in 1826 (Untersuchungen uber die Reihe 1 + (m/\)x + 2 ),
while Cauchy, in the early twenties in his Cours d' Analyse 9 undertook, for
the first time, a general discussion of the convergence of series. The result
of these investigations, for the series which we have under consideration,
is that all the earlier developments are sometimes correct, although the
rigorous proofs are very complicated. For the detailed consideration of
such proofs, in modern form, I refer you again toBurkhardt'sAlgebraische
Analysis or to Weber- Wellstein.

2. Although we shall have occasion to talk about it in detail later,
I must mention here the final foundation by Cauchy of the infinitesimal
calculus. By means of it the theory of the logarithm, which we discussed
above as taking its start at the hands of Biirgi and Napier in the seven-
teenth century, was established with full mathematical exactness.

3. Finally, we must mention the rise of that theory which is in-
dispensable to a complete understanding of the logarithmic and ex-
ponential functions, namely, the theory of functions of a complex
argument, often called, briefly, function thedry. Gauss was the first
to have a complete view of the foundations of this theory, even though
he published little or nothing concerning it. In a letter to Bessel, dated
December 18, 1811, but published much later 4 , he sketches and explains

with admirable clearness the significance of the integral / dz[z in the

complex plane, in so far as it is an infinitely many-valued function.
The fame of having also created independently the complex function
theory and of having made it known to the mathematical world belongs,
however, to Cauchy.

The result of these developments, insofar as it concerns our special
subject, might be briefly stated as follows: The introduction of the
logarithm by means of the quadrature of the hyperbola is the equal in
rigor of any other method, wheteas it surpasses all others, as we have
seen, in simplicity and clearness.

* Memoir on the hypergeometric series.

1 Commentationes societatis regiae Gottingiensis recentiores, vol. 11 (1813),
No. l, pp. 1 46. Werke vol. 3, pp. 123 162. German translation by Simon,
Berlin 1888.

2 Journal ftir Mathematik, vol. 1 (1826), pp. 311 339- Ostwalds Klassiker
No. 71.

3 Premiere Partie, Analyse Algdbrique. Paris 1821. = CEuvres, 2nd series, vol. 3,
Paris, 1897. German translation by Itzigsohn. Berlin 1885-

4 Briefwechsel zwischen Gauss unct Bessel, edited by Auwers. Berlin 1880;
or Gauss Werke, vol. 8 (1900), p. 90.

The Theory of Logarithms in the Schools. 155

3. The Theory of Logarithms in the Schools

It is remarkable that this modern development has passed over the
schools without having, for the most part, the slightest effect on the
instruction, an evil to which I have often alluded. The teacher manages
to get along still with the cumbersome algebraic analysis, in spite of
its difficulties and imperfections, and avoids the smooth infinitesimal
calculus, although the eighteenth century shyness toward it has long
lost all point. The reason for this probably lies in the fact that mathe-
matical instruction in the schools and the onward march of investigation
lost all touch with each other after the beginning of the nineteenth
century. And this is the more remarkable since the specific training
of future teachers of mathematics dates from the early decades of that
century. I called attention in the preface to this discontinuity, which
was of long standing, and which resisted every reform of the school
tradition : In the schools, namely, one cared little whether and how the
given theorems were extended at the university and one was therefore satis-
fied often with definitions which were perhaps sufficient for the present,
but which failed to meet later demands. In a word, Euler remained
the standard for the schools. And conversely, the university frequently
takes little trouble to make connection with what has been given in
the schools, but builds up its own system, sometimes dismissing this
or that with brief consideration and with the inappropriate remark:
"You had this at school".

In view of this, it is interesting to note that thpse university teachers
who give lectures to wider circles, e.g. to students of natural science
and technology, have, of their own accord, adopted a method of intro-
ducing the logarithm which is quite similar to the one which I am
recommending. Let me mention here, in particular, Scheffer's Lehrbuch
der Mathematik fur Studierende der Naturwissenschaften und Technik* 1 .
You will find there in chapters six and seven a very detailed theory
of the logarithm and the exponential function, which coincides entirely
with our plan and which is followed in chapter eight by a similar theory
of the trigonometric functions. I urge you to make the acquaintance
of this book. It is very appropriate for teachers, for whom it is designed,
in that the material is presented fully, in readable form, and adapted
to the comprehension even of the less gifted. Note, too, the great
pedagogic skill of Scheffers when he (to cite one example) continually
draws attention to the small number of formulas in the theory of
logarithms that one needs to know by heart, provided the subject is
once understood; for one can then easily look them up when they are
needed. In this way he encourages the reader to persevere in face of

* Textbook of Mathematics for Students of Natural Science and Technology.
1 Leipzig, 1905; fifth ed. 1921.

-JJ6 Analysis: Logarithmic and Exponential Functions.

the great mass of new material. I call your attention also to the fact
that although Scheffers takes it for granted that the subject has been
studied in school, he nevertheless develops it here in detail, on the
assumption that most of what was learned in school has been forgotten.
In spite of this, it does not occur to Scheffers to make proposals for a
reform of instruction in the schools, as I am doing.

I should like to outline briefly once more
my plan for introducing the logarithm into
the schools in this simple and natural way.
rj~i<>lL The first principle is that the proper source
~~~~ from which to bring in new functions is the

quadrature of known curves. This corre-
sponds, as I have shown, not only to the
historical situation but also to the procedure
in the higher fields of mathematics, e. g., in
elliptic functions. Following this principle

one would start with the hyperbola 77 = \ / and define the logarithm
of x as the area under this curve between the ordinates = 1 and = x
(see Fig. 57). If the end ordinate is allowed to vary, it is easy to see
how the area changes with and hence to draw approximately the
curve r] = log .

In order now to obtain simply the functional equation of the logarithm
we can start with the relation

/ x d^ f cx dl;
L T = Jo T'

which is obtained by applying the transformation c = ' to the variable
of integration. This means that the area between the ordinates 1 and x
is the same as that between the ordinates c and ex which are c times
as far from the origin. We can make this clear geometrically by ob-
serving that the area remains the same when we slide it along the
axis under the curve provided we stretch the width in the same ratio
as we shrink the height. From this the addition theorem follows at once :

Ji Ji A Jxi Ji

I wish very much that some one would give this plan a practical
test in the schools. Just how it should be carried out in detail must, of
course, be decided by the experienced school man. In the Meran school
curriculum we did not quite venture to propose this as the standard
method.

4. The Standpoint of Function Theory

Let us, finally, see how the modern theory of functions disposes of
the logarithm. We shall find that all the difficulties which we met in

The Standpoint of Function Theory.

our earlier discussion will be fully cleared away. From now on we shall
use, instead of y and x> the complex variables w = u + iv and z = x
+ iy. Then

1. The logarithm is defined by means of the integral

(1) w-

where the path of integration is any curve in the f plane joining ? = 1
to f = *.

2. The integral has infinitely many values according as the path
of integration encircles the origin 0,1,2,... times, so that log z is
an infinitely-many-valued function.

One definite value, the principal -Plane:

value [log z] , is determined if we

slit the plane along the negative real

axis and agree that the path of

integration shall not cross this cut.

It still remains arbitrary, of course,

whether we shall choose to reach

the negative real values from above Fi gi 5 s.

or from below. According to the

decision on this point the logarithm has + n i or ni for its imaginary

part. The general value of the logarithm is obtained from the principal

value by the addition of an arbitrary multiple of 2in\

(2) log* = [log z] + 2kni, (k = 0, 1 , 2, . . .) .

3. It follows from the integral definition of w = logz that the
inverse function z = f (w) satisfies the differential equation

From this we can at once write down the power series for /

it \ * i w . w * . w * i
, = /() = !+_ + _+_+....

Since this series converges for every finite w, we can infer that the
inverse function is a single-valued function which can be singular only
for w = oo, i.e., that it is an integral transcendental function.

4. The addition theorem for the logarithm is derived from the
integral definition, just as for real variables. From it we obtain for
the inverse function the equation

(4) /K)-/K) = /(^i + ^).

Similarly, it follows from (2) that

(5) f(w + 2kni) = f (w), (k = 0, 1, 2, . . .)

i.e., / (w) is a simply periodic function with the period 2 n i .

1J8 Analysis: Logarithmic and Exponential Functions.

5. If we put / (1) = e y it follows from (4) that for every rational

n t
value m/n of w the function / (w) will be one of the n values of y e m , as

this expression is usually defined; that is

We shall adopt the customary notation, and denote this one value of
f(w) by e w = e m/n , so that e w is a well defined single-valued function,
and indeed, the one given by equation (3).

6. What sort of a function, then, shall we understand, in the most
general sense, by the power b w with an arbitrary base 6? We must
adopt such conventions, of course, that the formal rules for exponents
are satisfied. In order then to establish a connection between b w and
the function e w which we have just defined, let us put b equal to e ] Kb ,
where log b has the infinitely many values

log b = [log 6] + 2 kni , (k = 0, 1 , 2, . . .)
It follow then that

b w = (e lo * b ) w = ^w- log ft e w[\oub] . 6 2kniv> 9 (& = 0, 1

and this expression represents, for the different values of k, infinitely
many functions which are completely unconnected. We have thus the
remarkable result that the values of the general exponential expression
b w , as these are obtained by the processes of raising to a power and
extracting a root, do not belong at all to one coherent analytic function,
but to infinitely many different functions of w , each of which is single-
valued.

The values of these functions are, to be sure, related to each other
in various ways. In particular they are all equal when w is an integer ;
and there are only a finite number of different ones among them
(namely, n) when w is a fraction mjn in its lowest terms. These n values
are ^m/ioiog* . g2fct(m/n) for A = 0, 1 , . . . , n - 1 , that is, the n values of

]/b m , as we should expect.

7. It is only now that we can appreciate the inappropriateness of
the traditional method which starts from involution and evolution and
expects to arrive at a single- valued exponential function. It finds itself
in an outright labyrinth in which it cannot possibly find its way by
so called elementary means, especially since it restricts itself to real
quantities. You will see this clearly if you will consider the situation
when 6 is negative, with the aid of the illuminating results which we
have just obtained. In this connection I merely remind you that we
are only now in a position to understand the suitableness of the definition
of the principal value (b > and b m/n > 0; see p. 145) which at the

The Standpoint of Function Theory.

time seemed arbitrary. It yields the values of one only of our infinitely
many functions, namely those of the function

[b w ] == M>[log6]

On the other hand, if n is even, the negative real values of b m ^ n will
constitute a set which is everywhere dense, but they belong to an
entirely different one of our infinitely many functions, and cannot
possible combine to form a continuous analytic curve.

I should now like to add a few remarks of a more serious nature
concerning the function theoretic nature of the logarithm. Since
w = log 2 suffers an increment of 2ni every time z makes a circuit
about z = 0, the corresponding Riemann surface of infinitely many
sheets must have at z = a branch point of infinitely high order so
that each circuit means a passage from one sheet into the next one.
If one goes over to the Riemann sphere it is easy to see that z = <x> is
another branch point of the same order and that there are no others.
We can now make clear what one calls the uniformizing power of the
logarithm of which we have already spoken in connection with the
solution of certain algebraic equations (see p. 133 e * sc l-)- To fix ideas
let us consider a rational power, z mfn . By reason of the relation

m m .

-logz

z n e n

this power will be a single-valued function of w = log z . This is expressed
by saying that it is uniformized by means of the logarithm. In order
to understand this, let us think of the Riemann surface of z min as well
as that of the logarithm, both spread over the z plane. This will have n
sheets and its branch points will also be at
2 = and z = oo, at each of which all the n
sheets will be cyclically connected. If we
now think of any closed path in the z plane
(see Fig. 59) along which the logarithm returns
to its initial value, which implies that its path
on the infinitely many sheeted surface is also
closed, it is easy to see that the image of this Fig. 59.

path will likewise be closed when it is mapped

upon the n sheeted surface. We infer from this geometric consideration
that z m/n will always return to its initial value when log z does, and
hence that it is a singlevalued function of log z. I am the more
willing to give this brief explanation because we have here the sim-
plest case of the principle of uniformization, which plays such an
important part in modern function theory.

We shall now try to make clearer the nature of the functional
relation w = log z by considering the conformal mapping upon the
w plane of the z plane and of the Riemann surface spread upon it. In
order not to be obliged to go back too far, let us refrain from including

160

Analysis: Logarithmic and Exponential Functions.

z-Plane:

w -Plane:

the corresponding spheres within the scope of our deliberations, in spite
of the fact that it would be preferable to do so. As before, we divide
the z plane along the axis of reals into a shaded (upper) and a unshaded
(lower) half plane. Each of these must have infinitely many images in
the w plane, since log z is infinitely many valued, and all these images
must lie in smooth connection with one another since the inverse function
z = e w is one valued. This means that the w plane is divided into
parallel strips of width n separated from one another by parallels to
the real axis (see Fig. 60). These strips are to be alternately shaded
and left blank (the first one above the real axis is shaded) and they
represent, accordingly, alternate conformal maps of the upper and lower

z half planes while the separating par-
allels correspond to the parts of the real
z axis. As to the correspondence in detail,
I shall remark only that z always appro-
aches when w , within a strip, tends to
the left toward infinity, that z becomes
infinite when w approaches infinity to
the right, and that the inverse function e w
has an essential singularity at w = oo.
I must not omit here to draw attention
to the connection between this represen-
tation and the theorem of Picard, since that
is one of the most interesting theorems of
the newer function theory. Let z (w) be
an integral transcendental function, that is, a function which has an
essential singularity only at w = oo (e.g. e w ). The question is whether
there can be values 2, and how many of them, which cannot be taken
at any finite value of w, but which are approached as a limit when w
becomes infinite in an appropriate way. The theorem of Picard states
that a function in the neighborhood of an essential singularity can omit
at most two different values; that an integral transcendental function,
therefore, can omit, besides 2 = oo, (which it of necessity omits), at
most one other value. e w is an example of a function which really
omits one other value besides oo, namely 2 = 0. In each of the parallel
strips of our division e w approaches each of these values but it assumes
neither of them for any finite value of w . The function sin w is an example
of a function which omits no value except 2 = oo.

I should like to conclude this discussion by bringing up again a
point which we have repeatedly touched and applying to it these
geometric aids. I refer to the passage to the limit from the power to
the exponential function which is given by the formula

-- SiJt-

Fig. 60.

The Standpoint of Function Theory.
If we put n w = v this takes the form

Let us, before passing to the limit, consider the function

/,(*)=(!+?)'.

whose function-theoretic behavior, as a power, is known to us. It has

a critical point, at w = v and w = oo, where the base becomes

and oo respectively, and it maps the / r half

planes conformally upon sectors of the w w .pi ane:

plane which have w = v as common vertex

and the angular opening n/v (see Fig. 61).

If v is not an integer this series of sectors

can cover the w plane a finite or an infinite

number of times, corresponding to the many

valuedness of /,, . If now v becomes infinite,

the vertex, v, of the sectors moves off Fig 61

without limit to the left and it is clear that

these sectors lying to the right of v go over into the parallel strips

of the w plane which belong to the limit function e w . This explains

geometrically the limit definition of e w . One can verify by calculation

that the width of the sectors at w = goes over _ .

w -Sphere?

into the strip width n of the parallel division.
But a doubt arises here. If v becomes infinite
continuously, it passes through, not only integral
but also rational and irrational values, for which
the f v will be many valued and will correspond to
many sheeted surfaces. How can these go over into
the smooth plane which corresponds to the single-
valued function e w ? If, for example, we allow v to
approach infinity only through rational values having
n for a denominator each /,, (w) will have an n sheeted Riemann surface.
In order to follow the limit process, let us, for a moment, consider
the w spKere. It is covered for each f v (w) with n sheets which are
connected at the branch points v and oo. Let the branch cut lie
along the minor meridian segment joining these points, as shown in
Fig. 62. If now v approaches oo the branch points coincide and the
branch cut disappears. Thus the bridge is destroyed that supplied the
connection between the sheets, there emerge n separate sheets and,
corresponding to them, n single-valued functions, of which only one is
our e w . If we now allow v to vary through all real values, we shall have,
in general, surfaces with infinitely many sheets whose connection is
broken in the limit. The values on one leaf of each of these surfaces

Klein, Elementary Mathematics. 1 1

162 Analysis: The Goniometric Functions.

converge toward the single-valued function e w t which is spread over the
smooth sphere, while the sequences of values on the other sheets have,
in general, no limit whatever. We thus have a complete explanation of
the right complicated and wonderful passage to the limit from the many
valued power to the single- valued exponential function.

As a general moral of these last considerations we might say that a
complete understanding of such problems is possible only when they
are taken into the field of complex numbers. Is this, then, not a sufficient
reason for teaching complex function theory in the schools? Max
Simon, for one, has in fact supported similar demands. I hardly believe,
however, that the average pupils, even in the highest class, can be
carried so far, and I think, therefore, that we should abandon those
aspects of method as to algebraic analysis in the schools which incline
toward such considerations, in favor of the simple and natural way which
we have developed above. I am, to be sure, all the more desirous that
the teacher shall be in full possession of all the function-theoretic
connections that come up here; for the teacher's knowledge should be
far greater than that which he presents to his pupils. He must be
familiar with the cliffs and the whirlpools in order to guide his pupils
safely past them.

After these detailed discussions we can now be briefer in the corres-
ponding consideration of the goniometric functions.

II. The Goniometric Functions

Let me say, before beginning, that the name goniometric functions
seems preferable to the customary name trigonometric functions, since
trigonometry is but a particular application of these functions, which
are of the greatest importance for mathematics as a whole. Their inverse
functions are analogous to the logarithm, while they themselves are
analogous to the exponential function. We shall call these inverse
functions the cyclometric functions.

1. Theory of the Goniometric Functions

As a starting point for our theoretical considerations let me suggest
the question as to the most appropriate way of introducing the gonio-
metric functions in the schools. I think that here also it would be best
to make use of our general principle of quadrature. The customary
procedure, which begins with the measurement of the circular arc, does
not seem to me to be so very obvious, and it lacks, above all, the ad-
vantage of affording a simple and coherent control both of elementary
and advanced fields.

Again I shall make immediate use of analytic geometry. Let us
start with the unit circle

Theory of the Goniometric Functions.

163

and consider the sector formed by the radii to the points A (x = 1 ,
y = o) and P (x, y) (see Fig. 63). In order to be in agreement with
the usual notation, I shall denote the area of this sector by 90/2 . (Then
the arc in the customary notation will be (p.)

I shall define the goniometric functions sine and cosine of q> as the
lengths of the coordinates x and y of the limiting point P of the sector 99/2 :

x = cos 9?,

y = sin 9? .

The origin of this notation is not clear. The word "sinus" probably arose

through an erroneous translation of an Arabic word into Latin. Since

we did not start from the arc we cannot

well designate the inverse functions, i. e.,

the double sector, as, a function of the

coordinates, by using the customary terms

arc sine and arc cosine, but it is natural

by analogy to call <p\2 the "area 11 of the

sine (or cosine) and to write

<p = 2 area siny = arc siny ,
9? = 2 area cosx = arc cos x .

Fig. 63.

The following notation, used in England and in America is also quite
appropriate :

cp = sin - I y .

= cos' 1 *,

The further goniometric functions:

, sin (p ,

tan w = - , ctno? =

r cos (p ' T

cos (jp

- - r -
sin (p

(in the older trigonometry also secant and cosecant) are defined as

simple rational combinations of the two fundamental functions. They

are introduced only with a view

to brevity in practical calcula- <**+- ^

tion and have for us no theo-

retical significance.

If we follow the coordinates
of P with increasing 9? we can
at once obtain qualitatively a
representation of the cosine and

sine curves in a rectangular coordinate system. They are the well
known wave lines with a certain period 2 n (see Fig. 64), where n is
defined as the area of the entire unit circle, instead of as usual, the
length of the semi-circle.

Let us now compare once more our introduction of the logarithm
and the exponential function with these definitions. You will recall that

11*

Fig. 64.

1(54 Analysis: The Goniometric Functions.

our point of departure was a rectangular hyperbola referred to its
asymptotes as axes.

-17 = 1.

The semi axis of this hyperbola is OA = ]/2 (see Fig. 65), whereas the
circle had the radius 1 . Let us now consider the area of the strip between
the fixed ordinate A A' (f = 1) and the variable ordinate PP'. If this

is called , we may put log I , and the
coordinates of P are expressed in terms of
in the form

You notice a certain analogy with the preceding
discussion, but that the analogy fails in two
respects. In the first place, is not a sector
as it was before, and furthermore the two coor-
dinates are now expressed rationally in terms of
one function e fp , whereas, in the case of the circle,

we had to introduce two functions, sine and cosine, to secure rational ex-
pressions. We shall see however that this divergence can be easily resolved.
Notice, in the first place, that the area of the triangle OP'P, namely
1? = , i s independent of the position of P. In particular, then,
it is the same as that of OA 'A . Therefore, if we add the latter triangle
to and then subtract the former triangle from this sum, we see that
can be defined as the area of a hyperbolic sector lying between a radius
vector to the vertex A and one to a variable point P , jiist as in the case
of the circle. There is still a difference in sign. Before, the arc AP,
looked at from , was counterclockwise, whereas now it is clockwise.
We can remove this difference by reflecting the hyperbola in OA , i.e.,
by interchanging and 77. We get then as coordinates of P

Finally let us introduce the principal axes in place of the asymptotes
as axes of reference, by turning Fig. 65 through 45 (after reflection
in OA). If we call the new coordinates (X , Y), the equations of this
transformation are

f2 1/2"

The equation of the hyperbola then becomes

and the sector now has precisely the same position that sector 0/2
had in the circle. The new coordinates of P as functions of may be
written in the form

j __ t + e 9 Y = e ~ e

Theory of the Goniometric Functions.

165

It remains only to reduce the entire figure in the ratio 1 : ]/2 in

order to make the semi axis of the hyperbola 1 instead of the ]/2, as
it was in the case of the circle. Then the sector in question has the area
<p/2, in complete accord with the pre-
ceding. If we call the new coordi-
nates (x, y) again, they will be the
following functions of

</> I rh

e ~r e

/y . ._

; which satisfy the relation

Fig. 66.

which is the equation of a hyperbola. These functions are called hyper-
bolic cosine and sine and are written in the form

x = cosh =

y = sinh =

The final result, then, is that if we treat the circle and the rectangular
hyperbola, each with semiaxis one, in literally the same way we obtain
on the one hand the ordinary goniometric functions, on the other the
hyperbolic functions, so that these functions correspond fully to one
another.

You know that these functions cosh and sinh can be used to ad-
vantage in many cases. Nevertheless we have really taken a step back-
ward here, so far as the treatment of the hyperbola is concerned. Whereas
at first, the coordinates ( , rj) could be rationally expressed in terms
of a single function e <t} ', it now requires two functions, which are connected
by an algebraic relation (the equation of the hyperbola). It is natural,
therefore to attempt a converse treatment for the goniometric functions,
analogous to the original developments for the hyperbola. This is, in
fact, quite easy if one does not object to the use of complex quantities,
and it leads to the setting up of a single fundamental function in terms
of which cos (p and sin y> can be expressed rationally, just as cosh <
and sinh & are in terms of e*, and which is therefore entitled to play
the chief role in the theory of the goniometric functions.

To this end we introduce into the equation of the circle x 2 + y 2 = 1
(where x = cos <p , y sin (p) the new coordinates

which gives

166

Analysis: The Goniometric Functions.

The desired central function is now the second coordinate vf 9 just
as it was above in the case of the hyperbola. If we denote it by / (99)
we have, by virtue of the equations of transformation:

i) = f(<p) = cos 99 + ishi99 , = TT-y = COS99 i sin 99 .
From the last equations we get

00599 =

sin a? = - ~ . - = ---
^ 2^

where we have complete analogy with the earlier relations between
cosh #, sinh<, and e . If prominence is thus given, from the start,
to the analogy between the circular and the hyperbolic functions, the
great discovery of Euler that / (<p) = e ltf) is divested of the mystery
that usually attaches to it.

The question now arises whether we cannot effect a similar reduction
of cos w and sin w to a single fundamental function, without leaving

the real field. This is indeed possible
if we look at our figures in the light
of project! ve geometry. In the case
of the hyperbola, in fact, we could
define the coordinate r\, which sup-
plied the fundamental function, as
parameter in a pencil of parallels
r] = constant. This means, projecti-
vely, so far as the hyperbola is con-
cerned, that we have a pencil of lines
with its vertex on the hyperbola (in
particular, here, at one of the infinitely
distant points)* If, now, in the case of either circle or hyperbola we
think of the parameter of any such pencil as a function of the area,
we obtain likewise a fundamental function and one which involves only
real quantities.

Let us think now of the circle (Fig. 67) and the pencil through the
point 5 (1,0)

Fig. 67.

where A is the parameter. On a former occasion (p. 45), we found as
the coordinates of the intersection P of the circle and the ray correspond-
ing to A,

1 - A 2 . 2A

x = cosy = y-j-jj- , y = sin 99 = j-^ .

so that

is, in fact, an appropriate real fundamental function. Moreover, since
Z PSO = i POA , and POA = q>, it follows at once that JL = tan y/2.

Theory of the Goniometric Functions. 167

The one-valued representation of sin <p and cos y in terms of tan <p/2
which appears in this way is often used in trigonometric calculations.
The connection between A and the earlier fundamental function f(<p)
appears from the last formula in the form

; = _?__ == 1 /-/"* = 1 _ / a - 1

* + 1 i "/ + /-i + 2 i /*'+"! + 2/

or conversely,

,, , . l - A 2

= * + i =

The introduction of A amounts, then, simply to the determination of a
linear fractional function of / (q>) which is real along the circumference
of the unit circle. In this way the formulas turn out to be real but
somewhat more complicated than by the immediate use of / (y) .

Whether one is willing to give up the advantage of reality in the
face of this disadvantage, depends, of course, upon how well the person
concerned knows how to deal with complex quantities. It is noteworthy,
in this connection, that physicists have long since gone over to the
use of complex quantities, especially in optics, for example, as soon
as they have to do with equations of vibration. Engineers, in particular
electrical engineers with their vector diagrams, have recently been
using complex quantities advantageously. We can say then that the
use of complex quantities is at last beginning to spread, even though
at present the great majority still prefer the restriction to real numbers.

Passing on to a brief survey of the farther development of the theory
of the goniometric functions, let us next consider certain fundamental
laws.

1 . The addition theorem for sin <p is

sin (cp -f- v) = sin 9? cos -^ +

and there is a corresponding formula for cos (<p + ^) . These formulas
appear to be more difficult than those for the exponential function,
due, of course, to the fact that we are not dealing here with the true
elementary function. This function, our / (9?) = cos q> + i sin <p, satis-
fies the very simple relation

v) = /(?)/(?).

which is precisely the formula for e tp .

2. It is easy now to obtain expressions for the functions of multiples
of an angle and of parts of an angle. Of these I shall mention only the
two formulas

cos <p <p

2 cos 2

because they were of such importance in constructing the first trigono-

168

Analysis: The Goniometric Functions.

metric tables. An elegant expression for all these relations is given by
De Moivre's formula

f(n-<p) =

where f(q>) = cos 9? + ism<p .

De Moivre, who was a Frenchman, but who lived in London, and was
in touch with Newton, published this formula in 1730 in his book
Miscellanea analytica.

3. From our original definition of y = sin <p, we can of course easily
derive an integral representation for the inverse q> = sin" 1 )/. The area
in Fig. 68, consisting of the sector <p/2 (A OP) of the unit circle, together
with the triangle OP'P , is bounded by the axes, a parallel to the x axis

at the distance y away, and the curve x = ]/l y 2 . Its area is there-

ry i --

fore / Vl y*dy. Since the triangle has the area
'

we have

V 1 - y 2 dy -

From this it follows by a simple transform-
ation that

cp = sin I y =

-T

Fig. 68.

We could proceed now just as in the case of
the logarithm, namely to develop the inte-
grand by the binomial theorem, and then to integrate term by term,
following Mercator. This would give us the power series for sin~ 1 y,
from which, by inversion, we could get the sine series itself. This is
the plan that Newton himself employed, as we have seen (p. 82).

4. I prefer, however, to take the shorter way which Taylor's great
discovery made possible. According to it one obtains from the above
integral formula the differential quotient for the sine itself

d y , o

+ I/ A yllZ _

dqp d

from which it follows that

dcosy

Taylor's theorem now gives

= sin <p ,

. +

> i I r |

9> 4
"4T

Theory of the Goniometric Functions.

169

It is easy to see that these series converge for every finite are therefore defined as single-
valued integral transcendental functions in the entire complex plane.
5. If we compare these series with the series for e v , we see that
the fundamental function / (<p) satisfies the relation

cos<p + i sin <p = e ir f .

This result is unambiguous because sin <p and cos (p as well as e <p are
single-valued integral functions.

6. It remains only to describe the nature of the complex functions
sin w , and cos w . We notice first that each of the inverse functions
w = sin~~ l z and w = cos"" 1 z yields a Riemann surface with an infinite

z-Plane:

number of leaves and with branch points at +1, 1, oo.

infinitely many branch points of the first order

lie over 2 = 4-! and z = \, while two

branch points of infinitely high order lie over

z = oo . In order to follow better the course

of the leaves in detail let us consider the divi-

sion of the w plane into regions which corre-

spond to the upper (shaded) and the lower

(unshaded) z half planes. For z = cosw this

division is brought about by the real axis and

by the parallels to the imaginary axis through

the points w = 0, n > 2 n, . . ., so that

the resulting triangular regions (see Fig. 69),

all extending to infinity, should be alternately

shaded and unshaded. At the points w = ,

In fact,

+ 7

Fig. 69.

ft , 4^, . . . (corresponding to z = +1), and at the points w =

> - (corresponding to z = 1) , four of the triangles meet. These
correspond to the four half leaves of the Riemann surface, which are
connected at each of the corresponding branch points lying above
z = -j-i f If w becomes infinite within any triangle, cos w approaches
the value z = oo. The fact that there are two separate sets of infinitely
many triangles each, all extending to infinity, corresponds to the situa-
tion that on the Riemann surface there are two separate sets of infinitely
many leaves connected at z = oo. For z = sin w the situation is
analogous, except that the representation in the w plane is moved to
the right by n/2. In these representations we find confirmation of my
earlier remarks (p. 1 60) concerning the nature of the essential singularity
at w = oo in its relation to the theorem of Picard.

2. Trigonometric Tables

After this brief survey of the theory of goniometric functions,
I wish to discuss something that is of prime importance in practical work,

-J70 Analysis: The Goniometric Functions.

namely trigonometric tables. At the same time I shall talk about loga-
rithmic tables, which I have thus far left in the background, for the
reason that from the beginning up to the present time the tabulation
of logarithms has gone hand in hand with that of trigonometric values.
The way in which logarithmic tables have reached their present form
is of extraordinary importance and interest for the mathematician in
the schools as well as in the university. I cannot describe in detail here,
of course, the long history of the development of such tables, but
I shall endeavor, by citing a few of the most significant works, to give
you a rough historical survey. Concerning other works, some of them
of equally great importance, which would round out the story, I refer
you to Tropfke or, so far as logarithmic tables are concerned, to the
exhaustive account in Mehmke's Encyclopedia report on numerisches
Rechnen (Enzyklopadie, I. F.), as well as to the French revision 1 of this
report by d'Ocagne.

I shall mention first the group of

A. Purely Trigonometric Tables

as they were developed before the invention of logarithms. Such tables
existed in ancient times, the first of which follows.

1 . The table of chords, by Ptolemy, which he compiled for astronomical
purposes about 150 A. D. This is to be found in his work Megale Syn-
taxis, in which he developed the astronomical system bearing his name,
and of which we have here a modern edition 2 . This work has come to
us, by way of the Arabs, under the much used title Almagest, which
is probably a combination of the Arabic article "al" with a mutilated
form of the Greek title. The table is constructed with thirty-minute
intervals. It does not give directly the sine of the angle #, but the chord
of its arc (i. e. 2 sin a/2) . The values of the chords are given in three
place sexagesimal fractions, that is in the form 0/60 + 6/3600 + c/216000,
where a,b,c are integers between and 59. The difficult thing for us,
however, is that these a , b , c are written, of course, in Greek number-
symbols, that is in combinations of Greek letters. The tables give also
the values of the differences, which permit one to interpolate fcr minutes.
In the calculation of his table, Ptolemy used, above all, the addition
theorem for trigonometric functions, in the form of the theorem on the
inscribed quadrilateral (Ptolemy's theorem). He used also the preceding
formula for sin <x/2 (i.e., the extraction of square root, in addition to
the rational operations), and he employed furthermore a process of inter-
polation.

1 Encyclopedic des Sciences Mathematiques, edition francaise, I, 23. See also
Cajori, F., History of Mathematics, 1919- Macmillan; and Smith, D. E., History 4
of Mathematics, 1925- Ginn.

2 Edited by Heiberg. 18981903. Leipzig.

Trigonometric Tables.

2. We advance now more than 1000 years to the time when tri-
gonometric tables were first made in Europe. The first person who
deserves mention is Regiomontanus (14361476), whose name was really
Johannes Miiller, but who changed it into the latinized form of Konigs-
berg, his birthplace. He calculated several trigonometric tables, in
which one sees distinctly the transition from the sexagesimal to the pure
decimal system. At that time no one thought of the trigonometric lines
as fractions corresponding to the radius one, as we do now. The values
were calculated for circles with very large radii, so that they appeared
as integers. To be sure, these large numbers were themselves written
as decimals, but in the choice of the radius one finds a persistent sug-
gestion of the sexagesimal system. Thus, in the first table of Regio-
montanus the radius is taken as 6000000, and not until he makes the
second table does he choose a pure decimal 10000000 and establish
complete accord with the decimal system. By the simple insertion of
a decimal point, the numbers of this table become decimals of today.
These tables of Regiomontanus were first published long after his death,
in the work of his teacher G. Peurbach: Tractatus super propositiones
Ptolemaei de sinubus et chordis 1 . Notice that this work, like so many
other fundamental works in mathematics*, was printed in Niirnberg in
the forties of the sixteenth century. Regiomontanus himself lived mostly
in Niirnberg.

3. I place before you now a work of the greatest general significance:
De revolutionibus orbium coelestium* by Nic. Copernicus, the book in
which the Copernican astronomical system is developed. Copernicus
lived from 1473 to 1543 in Thorn, but this work appeared likewise in
Niirnberg, two years after the publication of Regiomontanus 1 tables.
Inasmuch as Copernicus never saw these tables, he was obliged to
compute for himself the little table of sines which you find in his book
and which was needed to work out his theory.

4. These tables by no means met the needs of the astronomers, so
that we see a pupil and friend of Copernicus attempting soon a much
larger work. His name was Rhaticus, which again is a latinized form of
the name, of his birthplace (Vorarlberg). He lived from 1514 to 1576,
and was professor at Wittenberg. You must relate all these things to
the general historical background of the time. Thus we are in the age
of the Reformation when, as you know, Wittenberg and the free city
Niirnberg were centers of intellectual life. Gradually, however, during
the struggles of the Reformation, the center of gravity of the political
and intellectual life moved away from the cities and toward the courts
of the princes. Thus while everything heretofore had been printed in

1 Norimbergae, 1541.

* I have already mentioned Cardanus and Stifel and shall soon mention others.

2 Norimbergae, 1543-

172 Analysis: The Goniometric Functions.

Niirnberg, the great tables of Rhaticus now appeared under the patronage
of the Elector Palatine and bore therefore his name Opus Palatinum 1 .
They were printed shortly after the death of Rhaticus. They were
much more complete than the preceding tables, containing the values
of the trigonometric lines to ten plaes at intervals of ten minutes, with,
to be sure, a good many errors.

5. A new edition of this table, very much improved, was published
by Pitiscus of Griinberg in Silesia (1561 1613), chaplain of the Elector
Palatine. This Thesaurus Mathematicus 2 , again printed under princely
subsidy, contained the trigonometric numbers to fifteen places, at inter-
vals of ten minutes. The work was essentially freer from errors than
that of Rhaticus, and was more compendious.

We must bear in mind that all these tables were constructed, in the
main, with the aid solely of the half-angle formula, together with inter-
polation, for at that time the infinite series for sin x and cos % did not
exist. We can appreciate, then, the prodigious diligence and labor which
is represented in these great works.

B. Logarithmic-Trigonometric Tables

These tables were succeeded immediately by the development of the
second group, the logarithmic-trigonometric tables, and it is a re-
markable coincidence, the irony of history, one might say, that a
year after the tables of trigonometric lines had attained, with Pitiscus,
a certain completeness, the first logarithms appeared and rendered these
tables superfluous, in that from then on, instead of sine and cosine,
one used their logarithms. I have already mentioned the first logarithmic
tables, those of Napier.

1. Mirifici Logarilhmorum Canonis Descriptio of Napier, in 1614.
Napier had in mind, primarily, the facilitating of trigonometric cacula-
tion. Consequently he did not give the logarithms of the natural num-
bers, but only the seven-place logarithms of the trigonometric lines, at
intervals of one minute.

2. The actual construction of logarithmic tables in their present
form is due mainly to the Englishman Henry Briggs (15564630) who
was in touch with Napier. He recognized the great advantage that
logarithms with base ten would have for practical calculation, since they
would fit our decimal system better, and he introduced this base instead
of that of Napier as early as 1617 in his Logarithmorum Chilias Prima,
giving us the "artificial" or common logarithms which bear his name.
In order to calculate these logarithms, Briggs devised a series of inter-
esting methods which permitted the determination of each logarithm as
accurately as one chose. Briggs' second considerable book bore the title

1 Heidelbergae, 1596. 2 Francofurtii, 1613.

Trigonometric Tables. \j<i

Arithmetica logarithmica 1 . In it he tabulates the logarithms of the
natural numbers themselves instead of those of the angle ratios, as Napier
had done. To be sure, Briggs never finished his calculations. He gave
the logarithms of the integers only from 1 to 20000 and 90000 to 100000,
but to fourteen places. It is remarkable that precisely the oldest tables
give the most places, whereas now we are content, for most purposes,
with very few places. I shall come back to this later. Briggs also compiled
the common logarithms of the trigonometric lines to ten places with
ten jninute intervals in his Trigonometria Britannica-.

3. The gap in Briggs' table was filled by the Dutchman Adrian
Vlacq, mathematician, printer, and dealer in books, who lived inGouda
near Ley den. He issued a second edition of Briggs' book 3 , which con-
tained the logarithms of all integers from \ to 100000 but only to ten
places. We may consider this as the source of all our current tables
of logarithms of natural numbers.

Concerning the further development of tables, I can mention here
only in a general way the points in which advances were made in later
years as compared with the above mentioned early beginnings.

a) The first essential advance was in the theory. The logarithmic
series furnished, namely, an extremely useful new method for the calcula-
tion of logarithms. The compilers of the first tables knew nothing about
these series. As we have seen, Napier calculated his logarithms by means
of the difference equation, that is, by successive addition of A x/x, with
the further aid of interpolation. The important device of square root
extraction appeared with Briggs. He made use of the fact, which was
mentioned moreover by Napier in his Constructio (see p. 147), that one
knows log y<z b = \ (log a + log b) as soon as one knows the logarithms
of a and b. It is probable that Vlacq also calculated in this way.

b) Essential progress was made by a more suitable arrangement in
printing the tables, whereby it was made possible to combine more
material, in a clearer way, in a smaller space.

c) Above all, the correctness of the tables, was considerably increased
by a careful check of the older ones, thereby eliminating numerous
errors, especially in the last figures.

Among the large number of tables which thus appeared, I shall
mention only the most famous one.

4. This is the Thesaurus Logarithmorum Completus (Vollstandige
Sammlung grosserer logarithmisch-trigonometrischer Tafeln*), by the
Austrian artillery officer Vega, which appeared in Leipzig in 1 794. The
original is rare, but a photostatic reprint appeared in Florence in 1896.

1 Londini, 1624. 2 Goudae, 1633-

3 Briggs, H., Arithmetica Logarithmica. Editio secunda aucta per Adr. Vlacq,
Goudae, 1628.

* Complete collection of larger logarithmic trigonometric tables.

174 Analysis: The Goniometric Functions.

The Thesaurus contains ten place logarithms of the natural numbers,
and of the trigonometric lines, in an arrangement that has since become
typical. Thus you find there, e.g., the small difference tables for facili-
tating interpolation.

If we come down now to the nineteenth century, we notice. a far
reaching popularization of logarithms, due partly to the fact that they
were introduced into the schools in the twenties, but also to the fact
that they found more and more application in physical and technical
practice. At the same time we find a reduction in the number of places.
For the needs of the schools, as well as those of technical practice, were
better met by tables which were not too bulky, especially since three
or four places were sufficient for the requisite accuracy in nearly all
practical cases. To be sure, we still had, in my school days, seven-
place tables, the reason assigned being that the pupils would obtain in
this way an impression of the "majesty of numbers' 1 . Our minds today
are in general more utilitarian, and we use throughout two, three, or at
most five-place tables. I shall show you today three modern tables,
selected at random. One is a handy little four place table by Schubert 1 .
In it you will find all manner of devices, such as printing in two colors,
repetition above and below, on every page, of guiding quantities, and
the like, in order to exclude misunderstanding. The second is a modern
American table by Huntington 2 , which is still more cunningly arranged,
where, e.g., the leaves are provided with projections and indentations
to enable one to turn up at once the desired page. Finally, I am showing
you a slide-rule, which, as you know, is nothing else than a three-place
logarithmic table in the very convenient form of a mechanical calculator.
You are all familiar, certainly, with this instrument, which every engineer
nowadays has with him constantly.

We have riot yet reached the end of the development, but we can
see pretty clearly what its further direction will be. Of late, the cal-
culating machine, of which I talked earlier (see p. 17 et seq.), has been
coming into extensive use, and it makes logarithmic tables superfluous,
since it permits a much more rapid and reliable direct multiplication.
At present, however, this machine is so expensive that only large offices
can afford it. When it has become considerably cheaper, a new phase
of numerical calculation will be inaugurated. So far as goniometry is
concerned, the old tables of Pitiscus, which became old fashioned so
soon after birth, will then come into their own ; for they supply directly
the trigonometric ratios with which the calculating machine can operate
at once, thus avoiding the use of logarithms.

[ l Now Schubert -Haussner, Vierstellige Tafeln und Gegentafeln, Sammlung
Goschen, Leipzig, 191 7-]

2 Huntington, C. V., Four- Place Tables. Abridged edition, Cambridge, Mas-
sachusetts. 1907.

Applications of Goniometric Functions. 175

3. Applications of Goniometric Functions

It remains for me now to give you a survey of the application of gonio-
metric functions. I shall consider three fields

A. Trigonometry, which, indeed, furnished the occasion for inventing
the goniometric functions.

B. Mechanics, where, in particular, the theory of small oscillations
offers a wide field for applications.

C. Representation of periodic functions by means of trigonometric series,
which, as is well known, plays an important part in the greatest variety
of problems.

Let us turn at once to the first subject.

A. Trigonometry, in particular, spherical trigonometry

We are in the presence here of a very old science, which was in full
flower in ancient Egypt, where it was encouraged by the needs of two
important sciences. Geodesy required the theory of the plane triangle,
and astronomy needed that of the spherical triangle. For the history
of astronomy we have the voluminous monograph in A. v. Braun-
muhl's Vorlesungen fiber Geschichte der Trigonometric 1 . On the practical
side of trigonometry the most informative book is E. Hammer's: Lehr-
buch der ebenen und sphdrischen Trigonometric 2 ', on the theoretical side,
the second volume of the work I have often mentioned, the Enzyklopadie
der Elementarmathematik of Weber- Wellstein.

Within the limits of these lectures I cannot, of course, develop
systematically the whole subject of trigonometry. That would be a
matter for special study. Furthermore, practical trigonometry is given
full consideration here in Gottingen in the regular lectures on geodesy
and spherical astronomy. I should prefer to talk to you exclusively
about a very interesting chapter of theoretical trigonometry which, in
spite of its great age, cannot be regarded as closed, and which, on the
contrary, contains many still unsolved problems and questions, of relati-
vely elementary character, whose study would, I think, be rewarding.
I refer to spherical trigonometry. You will find this subject very fully
consideredin Weber-Wellstein, where importance is given to the thoughts
which Study developed in his fundamental work Spharische Trigono-
metric, orthogonale Substitutionen und elliptische Funktionen 3 . I shall try
to give you a survey of all the theories that belong here and to call
your attention to the questions which are still unanswered.

The elementary notion of a spherical triangle hardly needs explana-

1 Two volumes. Leipzig, 1900 and 1903-

2 Stuttgart, 1906. [Fifth edition, 1923-]

3 Abhandlungen der Mathematisch-physikalischen Klasse der Koniglich
Sachsischen Gesellschaft der Wissenschaften, vol. 20, No. 2. Leipzig, 1893-

176

Analysis: The Goniometric Functions.

tion. Three points on a sphere, no two of which are diametrially opposite,
determine uniquely a triangle in which each angle and each side lies
between and n (see Fig. 70). Further investigation discloses that it
is desirable to think of the sides and of the angles as unrestricted vari-
ables, which can thus be greater than n or 2 n , or multiples of these
values. One has to do then with sides that overlap and with angles
which wind multiply around their vertices. It becomes necessary there-
fore to adopt conventions concerning the signs of these quantities as
well as the sense in which they are measured. It is due to Mobius,
the great geometer of Leipzig, that the importance of the principle
of signs was consistently developed, and the way
opened for the general investigation of these quan-
tities under unrestricted variation. The part of
his work which is of particular significance here
is the Entwicklung der Grundformeln der spharischen
Trigonometric in grosstmoglicher Allgemeinheit 1 .

This determination of the sign begins with the
assumption of a definite sense of rotation about a
point A on a sphere in which the angle shall be
called positive (see Fig. 71). If this sense is settled for one point,
it is for every other point, since the first point can be moved con-
tinuously to that other. It is customary to select the counterclockwise
rotation as positive, whereby we think of ourselves as looking at the

sphere from the outside. Secondly, we must
assign a sense of direction to each great circle
on the sphere. We cannot be satisfied with an
initial determination for one great circle and
the continuous moving of it into coincidence
with any second great circle, because this coin-
cidence can be effected in two distinct ways.
On this account, we shall assign a sense of
direction separately to each great circle which
we consider, and we shall look upon one and
the same circle as, in a sense, two different configurations according
as we have assigned to it the one or the other direction. - With this
understanding, each directed great circle a can be uniquely related
to a pole P , namely to that one of its two poles, in the elementary
sense, from which its sense of direction would appear positive. Con-
versely, every point on the sphere has a unique polar circle with a
definite direction. With these considerations, the polarizing process, so
important in trigonometry, is uniquely determined.

Fig. 71.

1 Berichte liber die Verhandlungen der Koniglich Sachsischen Gesellschaft der
Wissenschaften, mathematisch-physikalische Klasse, vol. 12 (i860). Reprinted in
Mfibius, F., Gesammelte Werke, vol. 2, p. 71. Leipzig, 1886.

Spherical Trigonometry.

If now three points A , B , C on the sphere are given, we must still
make certain agreements, before a spherical triangle with these vertices
is uniquely dtermined. In the first place, the direction of each great
circle through A , B , C must be assigned, and we must know how many
revolutions are necessary in order to bring a point from B to C, from
C to A , and from A to B . The lengths a,b, c, determined in this way,
which may be arbitrary real quantities, are called sides of the spherical
triangle. Of course they are thought of as drawn on a sphere of radius
one. The angles are then defined as follows: oc is that rotation, about A
in positive sense, which would bring the direction CA into coincidence
with the direction A B , to which arbitrary multiples of ^ 2 n may be
added. The other angles are defined ana-
logously. If we now examine an ordinary
elementary triangle, as shown in Fig. 72,
and determine the directions of the sides
so that a , b, c are less than n , we find that
the angles a, /?, y are, according to our
new definiton, the exterior angles instead
of the interior angles as in the usual
consideration of the elementary triangle.

It has been known for a long while that by replacing the customary
angles of a spherical triangle by their supplements, in this manner, the
formulas of spherical trigonometry turn out to be more symmetrical
and perspicuous. The deeper reason for this appears from the following
consideration. The polarizing process described above, by virtue of the
conventions of Mobius, furnishes uniquely, for every given triangle,
another triangle called the polar triangle of the first; and it is easy to
see that, in view of our new definition, this polar triangle has for its
sides and angles the angles and sides, respectively, of the original tri-
angle. According to our agreements, then, every formula of spherical
trigonometry must still hold if we interchange in it a, b, and c with <x , P ,
and v , respectively, so that there must always be this simple symmetry.
If, on the other hand, the sides and angles are measured in the usual
way, this symmetry is lost ; for the relation between triangle and polar
triangle depends upon how one chooses the sides and angles in a given
case, and upon how one resolves the ambiguity of the pole in the case
of a non directed given circle.

It is clear now that, of the six parts of a spherical triangle defined
in this way, only three can be independent continuous variables, e.g. two
sides and the included angle. The formulas of spherical trigonometry
represent a number of relations between these parts or, to be more
exact, of algebraic relations between their twelve sines and cosines, in
which only three of these twelve magnitudes can be allowed to vary
arbitrarily, while the other nine depend algebraically upon them. If

Klein, Elementary Mathematics. 12

Analysis: The Goniometric Functions.

we go over to the sine and cosine, we can ignore the additive arbitrary
multiples of 2 n . Let us now think of trigonometry as the aggregate of
all possible such algebraic relations of this kind. Then we can state its
problem, according to the modern manner of thinking, as follows. If
we interpret the quantities

as coordinates in a twelve dimensional space R 12 , then the totality of
those of its points which correspond to actually possible spherical tri-
angles a , . . . , y represents a three-dimensional algebraic configuration
M 3 of this JR 12 , and the problem is to study this M 3 in the R IZ . In this
manner spherical trigonometry is coordinated with general analytic
geometry of hyperspace.

Now this M 3 must have various simple symmetries. Thus the
polarizing process showed that the interchange oia,b,c with <*, P,y,
always yielded a spherical triangle. Translated into our new language,
this states that when one interchanges x lt x 2 , x 3 , y lf y 2 , y 3 with # 4 , x 5 ,
x B>y*>y5>y6> respectively, any point of M 3 goes over into another
point belonging to it. Further, corresponding to the division of space
into eight octants by the planes of the three great circles, there exists
for any triangle seven auxiliary triangles whose parts arise from those
of the initial triangle through change of sign and the addition of n . This
yields for every point of M 3 seven further points whose coordinates
x lf . . ., XQ arise as a result of sign change. The totality of these sym-
metries leads to a certain group of substitutions and sign changes of
the coordinates of j?? 12 , which transforms M 3 into itself.

The most important question now is that concerning the algebraic
equations which are satisfied by the coordinates of M 3t and which
constitute the totality of trigonometric formulas. Since cos 2 a + sin 2 oc
= 1 , we have, to start with, the six quadratic relations

(1) *! + :v! = i, (1 = 1, 2, ...,6),

or, speaking geometrically, six cylindrical surfaces .F (2) of order two
passing through M 3 .

Six further formulas are supplied by the cosine theorem of spherical
trigonometry, which in our notation, is

cos a = cos & cose sin 6 sine cos a,
from which one gets by polarization

cos<x = cos/? cosy sm/Ssiny cosa .

These equations, together with the four others which arise through
cyclic permutation of a,b,c and ot, /?, y determine, all told, six cubic
surfaces F^ passing through M 3 :

(2) x l = x 2 x 3 - y 2 y 3 * 4 , * 2 = x 3 x l - y 3 y^x, , x 3 =

(3) * 4 = *s*6 -

Spherical Trigonometry.

Finally, we can make use of the sine theorem, which can be expressed
by the vanishing of the minors of the following matrix

sin a, sin 6, sine
sin a , sin/J, siny
or, written at length,

t A\ f\

These expressions represent three quadratic surface F* 9 of which only
two, to be sure, are independent. Thus we have set up altogether
fifteen equations for our M 3 in R 12 .

Now, in general, 12 3 =9 equations do not, by any means,
suffice to determine a three dimensional algebraic configuration in R 12 .
Even in the ordinary geometry of R z , not every space curve can be
represented as the complete intersection of two algebraic surfaces. The
simplest example here is the space curve of order three which requires
for its determination at least three equations. It is easy to see that,
in our case also, the nine equations (1) and (2) do not determine M 3 .
It is well known, namely, that the sine theorem can be derived from
the cosine theorem only to within the sign, which one then determines,
ordinarily, by geometric considerations. We should like to know then
how many, and which, of the trigonometric equations really determine
our M 3 completely. In this connection I should like to formulate four
definite questions to which the literature thus far appears to give no
precise answer. It could be a worth-while task to investigate them thor-
oughly. That would probably not be especially difficult, after one had
acquired a certain skill in handling the formulas of spherical trigono-
metry. My questions are:

1. What is the order of M 3 ?

2. What are the equations of lowest degree by means of which M 3
can be completely represented?

3. What is the complete system of linearly independent equations
which represent M 3 , i.e., of equations / a = 0, . . ., f n = such that the
equation of every other surface passing through M 3 could be written
in the forih m^^ + . . . + m n f n = 0, where m lf . . ., m n are integers?
It is possible that more equations may be needed here than in 2.

4. What algebraic identities (so called syzygies) exist between these
formulas f lf . . ., / n ?

One could gain familiarity with these things by consulting in-
vestigations which have been made in exactly the same direction but
in which the questions have been put somewhat differently. These
appear in the Gottingen dissertation 1 , 1894, of Miss Chisholm (now

1 Algebraisch-gruppentheoretische Untersuchungen zur sphdrischen Trigono-
metrie, Gottingen, 1895-

12*

Analysis: The Goniometric Functions.

Mrs. Young), who, by the way, was the first woman to pass the normal
examination in Prussia for the doctor's degree. The most noteworthy
of Miss Chisholm's various preliminary assumptions is her selection of
the cotangents of the half angles and sides as independent coordinates.
Since tan (a/2) and likewise, of course, ctn (a/2), is a fundamental
function, in terms of which sin a and cos a can be uniquely expressed,
it is possible to write all the trigonometric equations as algebraic relations
between ctn (a/2) , . . . , ctn (y/2) . The spherical triangles constitute now
a three-dimensional configuration M 3 in a six dimensional space R Q
which has ctn (a/2), . . ., ctn (c/2), ctn (a/2), . . ., ctn (y/2), as coordi-
nates. Miss Chisholm shows that this M 3 is of order eight and that it
can be fully represented as the complete intersection of three surfaces
of degree two (quadratic equations) of R 6 ] and she investigates also
the questions which arise here, which are analogous to those stated
above.

In my lectures on the hypergeometric function 1 , I called the group
of formulas of spherical trigonometry which I have discussed above,
and which connect the sines and the cosines of the sides and angles,
formulas of the first kind, in distinction from an essentially different
group of formulas which I called formulas of the second kind. The latter
are algebraic equations between the trigonometric functions of the half
angles and sides. In studying them it will be best to select the twelve
quantities

a . a a . a

cos , smy , . . .; cos ~ 2 - > sm 2

as coordinates in a new twelve space R\% , in which the spherical triangles
again constitute a three-dimensional configuration M'% . It is here that
those elegant formulas appear which, at the beginning of the last
century, were published independently and almost simultaneously by
Delambre (1807), Mollweide (1808) and finally Gauss 1809 [in the Theoria
motus corporum coelestium, No. 54 2 ]. These are twelve formulas which
arise by cyclic permutation in:

8 -\- y b c .ft v b c

sin^p cos- sm sin'

2,2 2 -r- ^

. a -^ a ' . <x a

sin cos sin sin-

2 2 2 ^

cos-

oc ' a ' a - a

cos-- cos cos- o sm---

2 2 2 &

1 Winter semester 1893 1894. Elaborated by E. Ritter. Reprinted Leipzig,
1906.

2 Reprinted in Werke, Leipzig, 1906, vol. 7, p. 67-

Spherical Trigonometry.

That which is essential and new in them, as opposed to the formulas
of the first kind, is the double sign, with respect to which the following
is true. For one and the same triangle, the same sign, either the upper
or the lower, holds for all twelve formulas, and there are triangles of
both sorts. The Mg of spherical triangles in the above defined R' 12 satis-
fies, in other words, two entirely different systems of twelve cubic
equations each, and divides therefore into two separate algebraic con-
figurations M 3 , for which the one sign holds, and M 3 , for which the
other holds. By virtue of this remarkable fact these formulas take on
the greatest significance for the theory of spherical triangles. They are
much more than mere transformations of the old equations which might
at most serve to facilitate trigonometric calculation. To be sure, De-
lambre and Mollweidc did consider these formulas only from this practical
standpoint. It was Gauss who had the deeper insight, for he draws
attention to the possibility of a change of sign "if one grasps in its
greatest generality the idea of spherical triangle". It seems to me
proper, therefore, that the formulas should bear Gauss's name, even if
he did not have priority of publication.

It was Study who first recognized the full range of this phenomenon,
and who developed it in his memoir of 1893, which I mentioned on p. 1 75.
His chief result can be stated most conveniently if we consider the six
space R Q which has for coordinates the quantities # , & , c , ft , /? , y them-
selves, thought of as unrestricted variables. I call them transcendental
parts of the triangle in destinction from the algebraic parts cos #,...,
or cos (a/2) , . . . , because the former arc transcendental functions, while
the latter are algebraic functions of the ordinary space coordinates of
the vertices of the triangle. In this JR 6 , the aggregate of all spherical
triangles appears as the transcendental configuration M ( > whose image
in R\2 is the algebraic M' 3 considered above. Since however the latter
split into two parts and the mapping functions cos (a/2) , . . . are single
valued continuous functions of the transcendental coordinates, the trans-
cendental M^ must split into at least two separated parts. Study's
theorem is as follows : The transcendental configuration M^ of the quan-
tities a,b,.c, <*, P,y, belonging to a spherical triangle of the most general
sort, divides into two separate parts corresponding to the double sign in
the Gaussian formulas, and each of these parts is a connected continuum.
The essential thing here is the exclusion of any farther division. It
would not be possible, by farther manipulation of the trigonometric
formulas, to bring about similar and equally significant groupings of
spherical triangles. The triangles of the first of these parts, that corre-
sponding to the upper sign in the Gaussian formulas, are called proper
triangles, those of the other, improper, and we may state Study's
theorem briefly as follows : The totality of all spherical triangles resolves
itself into a continuum of proper and one of improper triangles. You

Analysis: The Goniometric Functions.

will find further details, and a proof of this theorem, in Weber-
Wellstein 1 . I am attempting here only to state the results clearly.
I must now say something further concerning the difference between
the two sorts of triangles. If a spherical triangle is given, i.e., an ad-
missible set of values of#,&,c,<x,/?,y, whose cosines and sines satisfy
the formulas of the first sort, and which therefore represents a point
of M^, how can we decide whether the triangle is proper or improper?
In order to answer this question we first find the smallest positive
residues a ,b ,c , <x , /3 , y of the given numbers, with respect to the
modulus 2n\

# = a (mod 2n) , . . . , <X Q = oc (mod 2n) , . . .
Q^a Q <27i, . . . , 0^<x < 2rc, . . .

Their sines and cosines coincide with those of a , . . . , a , . . . so that
they also represent a triangle which we shall call the reduced, or the
Moebius, triangle corresponding to the given one, since Moebius himself
did not consider the parts as varying beyond 2 n . Then we can deter-
mine, by means of a table, whether the Moebius triangle is proper or
improper. You will find this, in a form somewhat less clear, in Weber-
Wellstein (p. 352, 379, 380), as well as figures (p. 348, 349) of the types
of proper and improper triangles. As is usual, I shall call an angle
reentrant when it lies between n and 2 n and I shall, for the sake of
brevity, apply this term also to the sides of the spherical triangle. Then
there are, altogether, four typical cases of each sort.

I. Proper Moebius triangles'.

1. sides reentrant; angles reentrant.

2. 1 side reentrant; 2 adjacent angles reentrant.

3. 2 sides reentrant; 1 included angle reentrant.

4. 3 sides reentrant; 3 angles reentrant.

II. Improper Moebius triangles'.

1. sides reentrant; 3 angles reentrant.

2. 1 side reentrant; 1 opposite angle reentrant.

3. 2 sides reentrant; 2 opposite angles reentrant.

4. 3 sides reentrant; angles reentrant.

There are no cases other than these, so that this table enables us actually
to determine the character of a Moebius triangle.

The transition to the general triangle a , . . . , oc, t . . . from the cor-
responding reduced triangle is made, after what was said above, by
means of the formulas:

a = + n^ 2n , b = b Q + n 2 *2n , c = C Q + n% 2n ,

* = #o + v \ ' 2n > P = A) + V 2 ' 2n > y = 7o + V 3 ' 2n
1 Vol. 2, second edition (1907), p. 385 ( 47).

Spherical Trigonometry.

We may then make use of the following theorem The character of the
general triangle is the same as or the reverse of that of the reduced triangle
according as the sum of the six integers n + n 2 + n 3 + v l -f- ^ 2 + V 3 ^ s
even or odd. Thus the character of every triangle as proper or improper
can be determined.

I shall conclude this chapter with a few remarks about the area
of spherical triangles. Nothing is said about this in Study or in Weber-
Wellstein. It does come up for consideration in my Alteren funktionen-
theoretischen Untersuchungen uber Kreisbogenfireiecke* . Up to this point
we have considered the triangle merely as an aggregate of three angles
and three sides which satisfy the sine and consine laws. In my in-
vestigations I was concerned with a definite area bounded by these
sides, in a certain sense with a membrane stretched between these
sides and involving appropriate angles.

Of course we can now no longer think of <x , ft , y as the exterior
angles of the triangle, as we did before for reasons of symmetry. We
shall talk, rather, of those angles which the membrane itself forms at
the vertices, and I shall call them interior angles of the triangle. I shall
denote them, as is my habit, by ATT, //TT, vn (see Fig. 73). These angles
can also be thought of as unrestricted positive variables, since the
membrane might wind about the vertices.
In accordance with this, I shall denote the
absolute lengths of the sides by In , mn , n n ,
which are also unrestricted positive variab-
les. But it will be no longer possible for
the sides and the angles to "overlap" in-
dependently of one another, i.e., to contain
arbitrary multiples of 2 n , as they could
before, for the fact that a singly-connected F te- 73.

membrane should exist with these sides

and angles finds its expression in certain relations between the numbers
of these overlappings. In my memoir Uber die Nullstellen der hyper-
geometrischen Reihe 1 I called these supplementary relations of spherical
trigonometry. If we denote by E (x) the largest positive integer which x
exceeds, [E (x) < x] , these relations are

* Earlier function-theoretic investigations of spherical triangles.
1 Mathematische Annalen, vol. 37 (1888). [Reprinted in Klein, F., Gesammelte
Mathematische Abhandlungen, vol. 2 (1921), p. 550.

Analysis: The Goniometric Functions.

and since E (1/2) , for example, gives the multiple of 2 n which is contained
in the side / n t these relations determine precisely the desired "overlap* '
numbers of the sides ln,mn t nn when one knows the angles AJT, ^n y vn
together with their overlap numbers. It is easy to see, in particular,
that of the three numbers A // v , l + p v, I p + v, one
at most can be positive. Consequently only one of the three arguments
on the right sides can exceed unity, and since E (x) = f or x ^ \ , it
is possible for only one of the overlap numbers to be different from zero.
In other words only one side, at most, of a triangular membrane can
overlap (be greater than 2) and that side must be opposite the largest
angle.

For the proof of these supplementary relations I refer you to my
mimeographed lectures Uber die hypergeometrische Funktion 1 (p. 384),
although the edition is long since exhausted. There, as well as in my
memoir in volume 37 of the Mathematische Annalen, the initial assump-
tions were somewhat broader than the present ones, in that spherical

triangles were considered which are bounded
by arbitrary circles on the sphere, not neces-
sarily by great circles. I shall sketch briefly
the train of thought of the proof. We start
with an elementary triangle, in which a
membrane can certainly be stretched, and
obtain from it step by step the most general
admissible triangular membrane by repe-
, atedly attaching circular membranes, either

at the sides, or, with branchpoints, at the
Fig . 74 . vertices. Fig. 74 shows, as an example, (in

stereographic projection) a triangle ABC

which arises from an elementary triangle by attaching the hemi-
sphere which is bounded by the great circle AB, whereby the side AB
overlaps as well as the angle C. It is clear that the supplementary
relations continue to hold here, and one sees in the same way that
they retain their validity for the most general triangular membrane
which can be built up by this process.

We must now inquire how these triangles, which satisfy the supple-
mentary relations, fit the general theory which we have discussed
already. They are obviously only special cases, (because the overlap
numbers of the sides and angles are, in general, entirely arbitrary)
special cases which are characterized by the possibility of framing a
stretched membrane in a triangle. At first one can really be puzzled
here, for we have seen that the totality of all proper triangles (some of
which do not need to satisfy the supplementary relations) constitutes

1 These lectures were referred to on p. 180.

Spherical Trigonometry.

185

Fig. 76.

a continuum, and that any one of them could be derived, therefore,
from an elementary triangle by a continuous deformation. One would
think, naturally, that it would be impossible, during this deformation
to lose the membrane which was stretched in the initial elementary
triangle. The explanation of this difficulty appears if we extend Moebius'
principle of sign-change to areas, by agreeing that an area is to be called
positive or negative according as its boundary is traversed in the positive
(counter clockwise) or negative sense. Accordingly, when a curve which
crosses itself bounds several partial areas, the entire area is the algebraic
sum of the several parts, each of these determined, as to sign, by the
sense in which its boundary is traversed. In Fig. 75 this would be the
difference, in Fig. 76, the sum of the parts which are distinguished by
different shading. These agreements are, of
course, merely the geometric expression of that
which the analytic definition itself supplies.

If we apply this, in particiilar, to triangles
formed by circular
arcs, it turns out, in
fact, that with every
proper triangle we can
associate an area on
the sphere such that, when one circuit of the triangle is made,
different parts of this area are combined with different signs
because the boundaries of these parts are traversed in different senses.
Those triangles for which the supplementary
relations hold are special, then, in that their
areas consist of a single piece of membrane
bounded by a positive circuit. It is this pro-
perty which gives them their great significance
for the function-theoretic purposes to which
I put them in my earlier studies.

I will now illustrate this situation by means
of an example. Let us consider the triangle
ABC in t stereographic projection (Fig. 77)
where, of the points of intersection A , A' of
the great circles BA, CA, A is the one more
remote from the arc B C . If we now transfer the
general definition of the exterior angles (p. 177) to their supplements, the
interior angles, we find that [An and vn measure the rotation of J5Cinto
BA and of CA into CB, respectively, and are, therefore, positive in our
case. Similarly kn measures the rotation of AB into AC and is there-
fore negative. Put A = - A', K > 0. Then the triangle A'BC is ob-
viously an elementary triangle with angles A'rc, ^n t vn , all of which
are positive. If we now make a circuit about the triangle ABC , the

Fig 77-

186 Analysis: The Goniometric Functions.

boundary of the elementary triangle A'BC will be traversed in the
positive sense but that of the spherical sector A A' in the negative, and
the area of the triangle ABC , in the Moebius sense, will be the difference
of these two areas. This breaking up of the triangular membrane into
a positive and a negative part can be visualized, perhaps, by supposing
the membrane twisted at A' so that the rear or negative side of the
sector is brought to the front. It is not hard to construct more difficult
examples after this pattern.

I shall now show, by means of this same example, that with this
general definition of area, the formulas for the area of elementary tri-
angles still remain valid. As you know, the area of a spherical triangle
with angles kn t [JLTI, vn t on a sphere with radius one, is given by the so-
called spherical excess (A -f // + v 1) n where A, /j, v > 0. Let us
now see that this formula holds also for the above triangle ABC. It
is clear that the area of the elementary triangle A 'B C is (A' -f // + v \ } n.
From this we must subtract the area of the sector A A' whose angle
is A'TT. But this is 2 I'M, because the area of a sector is proportional to
its angle; and it becomes 4n when the angle is 2 n (the entire sphere).
We get then, as the area of ABC,

(X + p + v \}n-2Vn = (V + /LL + V \)n= (i + p + v \)n.

It is probable, if we had a general proper triangle with arbitrary sides
and angles, and if we should try to fit into it a multi-parted membrane
and determine its area (which, according to the sign rule, would be
the algebraic sum of the parts), that the result would show the general
validity of the formula (A -}- /j + i> \)n, where, of course, Arc, . . .
are the real angles of the membrane, and not, as before, the exterior
angles. The investigation suggested here has not been carried out,
however. It would certainly not offer great difficulties, and I should
be glad if it were undertaken. At the same time, it would be important
to determine, from the present standpoint, the role of the improper
triangles.

With this I shall leave the subject of trigonometry and go over to
the second important application of goniometric functions, one which
also falls within the field of the schools.

B. Theory of small oscillations, especially those of the

pendulum

I shall recall briefly the deduction of the law of the pendulum as we
are in the habit of giving it at the university, by means of infinitesimal
calculus. A pendulum (see Fig. 78) of mass m hangs by a thread of
length /, its angle of deflection from the normal being <p. Since the
force of gravity acts vertically downwards, it follows from the funda-

Small Oscillations. jgj

mental laws of mechanics that the motion of the pendulum is deter-
mined by the equation

(1) .

For small amplitudes we may replace sin <p by <p without serious error.
This gives for so called infinitely small oscillation of the pendulum

The general integral of this differential equa-
tion is given, as you know, by goniometric func-
tions, which are important here, as I said before.
precisely by reason of their differential properties
The general integral is

where A , B are arbitrary constants. If we introduce appropriate new
constants C,/ , we find

(3) p = C

where C is called the amplitude and / the phase of the oscillation.
From this we get, for the duration of a complete oscillation, T 2n]/l/g.

Now these are very simple and clear considerations, and if we went
more fully into the subject they could of course be given graphical
form. But how different they appear from the so called elementary
treatment of the pendulum law which is widely used in school instruction.
In this, one endeavors, at all costs, to avoid a consistent use of infinitesi-
mal calculus, although it is precisely here that the essential nature of
the problem demands emphatically the application of infinitesimal
methods. Thus one uses methods contrived ad hoc, which involve
infinitesimal notions without calling them by their right name. Such
a plan is, of course, extremely complicated, if it is to be at all exact.
Consequently it is often presented in a manner so incomplete that it
cannot be thought of, for a moment, as a proof of the pendulum law.
Then we have the curious phenomenon that one and the same teacher,
during one hour, the one devoted to mathematics, makes the very
highest demands as to the logical exactness of all conclusions. In his
judgment, still steeped in the traditions of the eighteenth century, his
demands are not satisfied by the infinitesimal calculus. In the next
hour, however, that devoted to physics, he accepts the most questionable
conclusions and makes the most daring application of infinitesimals.

To make this clearer, let me give, briefly, the train of thought of
such an elementary deduction of the pendulum law, one which is actually
found in text books and used in instruction. One begins with a canonical

-jgg Analysis: The Goniometric Functions.

pendulum, i.e. a pendulum in space whose point moves with uniform
velocity v in a circle about the vertical, as axis, so that the suspending
thread describes a circular cone (see Fig. 79). This is the motion which
is called in mechanics regular precession. The possibility of such motion
is, of course, assumed in the schools as a datum of experience and the
question is asked merely concerning the relation which obtains between
the velocity v and the constant deflection of the pendulum, cp = oc (angular
opening of the cone which is described by the thread).

One notices, first, that the point of the pendulum describes a circle
of radius r = I sin oc , for which one may write r = I oc when oc is

sufficiently small. Then one talks of cen-
trifugal force and reasons that the point,
with mass m, revolving with velocity v ,
must exert the centrifugal force

z; 2 v 2

m = m -,
r I - a

In order to maintain the motion there
must be an equal centripetal force directed
toward the center of the circular path.
Fig. 79. This is found by resolving the force of

gravity into two components, one directed

along the thread of the pendulum, the other, the desired force, acting
in the plane of the circular path and directed toward its center, having
the magnitude m g tan oc (see Fig. 79). This can be replaced by mg - oc
when oc is sufficiently small. We obtain, then, the desired relation in
the form

m -- = mg oc , or v = oc 1/g / .
lot, '

The time of oscillation T of the pendulum, that is, the time in which
the entire circumference of the circle 2nr = 2nloc is traversed, is then

= 2

In other words, when the angle of oscillation oc is sufficiently small,
the canonical pendulum performs a regular precession in this time,
which is independent of oc.

To criticize briefly this part of the deduction, we might admit the
validity of replacing sin oc and tan oc by oc itself, which we did ourselves
in our exact deduction (p. 187); for this permits the transition from
"finite" to "infinitely small" oscillations. On the other hand, we must
call attention to the fact that the formula used above for centrifugal
force can be deduced in "elementary" fashion only by neglecting all
sorts of small quantities; and the exact justification for this is founded
precisely on differential calculus. The very definition of centrifugal

Small Oscillations.

189

force, for example, requires in fact the notion of the second differential
coefficient, so that the elementary deduction must also smuggle this in.
And since in doing this, one is unable to say clearly and precisely what
one is talking about, there arise the greatest obstacles to understanding,
which are not present at all when the differential calculus is used. I do
not need to go into detail here because I can refer you to some very
readable articles on school programs, by the deceased realgymnasium
director H. Seeger 1 , in Gustrow and to a very interesting study by
H. E. Timerding: Die Mathematik in den
physikalischen Lehrbuchern 2 . In Seeger you
will find, among other things, an exhaustive
criticism of the deductions of the formula
for centrifugal force, in a manner corre-
sponding to our standpoint. In Timerding
there are extensive studies of the mathe-
matical methods which are traditionally used
in the teaching of physics*. Let me now
continue with the discussion of pendulum
oscillations.

The considerations set forth above show the possibility of uniform
motion in a circle. If we now set up an x y coordinate system (see
Fig. 80) in the plane of this circle (i.e., in view of our approximation,
the tangent plane to the sphere), this motion will, in the language of
analytic mechanics, be given by the equations

(4)

*-/* cos J/|-(/-g

y = / * sin J/| (t - t Q )

But we wish the plane oscillations of the pendulum; that is, the
point of the pendulum in our x y plane is to move on a straight line,
the x axis. The equations of its motion must be

(5)

= 0,

1 Vber die Stellung des hiesigen Realgymnasiums zu einem Beschlusse der letzten
Berliner Schulkonferenz (Gustrow, 1891, Schulprogramm No. 649). Vber die Stellung
.des hiesigen Realgymnasiums zu dem Erlass des preussischen Unterrichtsministeriums
von 1892 (1893, No. 653)- Bemerhungen uber Abgrenzung und Verwertung des
Unterrichts in den Elementen der Infinitesimalrechnung (1894, No. 658).

2 Bd. Ill, Heft 2 der "Abhandlungen des deutschen Unterausschusses der
Intern ationalen mathematischen Unterrichtskom mission' 1 . Leipzig u. Berlin 1910.

* See also Report on the Correlation of Mathematics and Science Teaching
by a joint committee of the British Mathematical Association and the Science
Masters Association 1908. Reprinted 1917. Bell and Sons, London.

Analysis: The Goniometric Functions.

in order that the correct equation (3) shall result when <p = x/l. Thus
we must pass from equations (4) to (5) without, however, making use
of the dynamical differential equations. This is made possible by setting
up the principle of superposition of small oscillations, according to which
the motion x + #i> y + y\ is possible when the motions x, y and x lf y l
are given. We may combine, namely, the counterclockwise pendulum
motion (4) with the clockwise motion

Xl = I . oc cos j/^ (t g , y l = Z (x sin j/ 1- (t t ) .

Then, if we put a = C/2, the motion x + x lt y + y l is precisely the
oscillating motion (5) which was desired.

In criticizing what precedes, we inquire, above all, how the principle
of superposition is to be established, or at least made plausible, without
the differential calculus. With these elementary presentations there
remains always the doubt as to whether or not our neglecting of suc-
cessive small quantities may not finally accumulate to a noticeable error,
even if each is permissible singly. But I do not need to carry this out
in detail, for these questions are so thoroughly elementary that each
of you can think them through when you feel so inclined. Let me, in
conclusion, state with emphasis that we are concerned in this whole
discussion with a central point in the problem of instruction. First,
the need for considering the infinitesimal calculus is evident. Moreover,
it is clear that we need also a general introduction of the goniometric
functions, independently of the geometry of the triangle, as a preparation
for such general applications.

I come now to the last of the applications of the goniometric functions
which I shall mention.

C. Representation of periodic functions by means of series
of goniometric functions (trigonometric series)

As you know, there is frequent occasion in astronomy, in mathe-
matical physics, etc., to consider periodic functions, and employ them
in calculation. The method indicated in the title is the most important

and the one most frequently
used. For convenience we
shall suppose the unit so
chosen that the given pe-
riodic function y / (x)

Fig. si. has the period 2n (see

Fig. 81). The question then

arises as to whether or not we can approximate to this function by
means of a sum of cosines and sines of integral multiples of x, from
the first, to the second,..., in general to the w-th, each, with a

Trigonometric Series.

properly chosen constant factor. In other words, can one replace f(x),
to within a sufficiently small error, by an expression of the form

S n (x) = ~ + a 1 cos x + a 2 cos 2x + + a n cosnx

+ b l sinx + b 2 s\n2x + + b n sinnx.

The factor \ is added to the constant term to enable us to give a general
expression for the coefficients.

First I must again complain about the presentation in the text books,
this time the texts in differential and integral calculus. Instead of
putting into the foreground the elementary problem which I have
outlined above, they often seem to think that the only problem which
is of any interest at all is the theoretical question, connected with the
one we have raised, whether / (x) can be exactly represented by an
infinite series. A notable exception to this is Runge in his Theorie und
Praxis der Reihen 1 . As a matter of fact, that theoretical question is,
in itself, thoroughly uninteresting for practical purposes, since we are
concerned in practice with a finite number of terms, and not too many
at that. Moreover it does not even permit a conclusion a posteriori as
to the practical usableness of the series. One may by no means conclude
from the convergence of a series that its first few terms afford even a
fair approximation to the sum. Conversely, the first few terms of a
divergent series may be useful, under certain conditions, in representing
a function. I am emphasizing these things because a person who knows
only the usual presentation and who wishes then to use finite trigono-
metric series in, say, the physical laboratory, is apt to be deceived
and to reach conclusions that are unsatisfactory.

The customary neglect of finite trigonometric sums seems still more
remarkable when one recalls that they have long been completely treated.
The astronomer Bessel gave the authoritative treatment in 1815- You
will find details concerning the history and literature of these questions
in the encyclopedia reference by Burkhardt on trigonometrische Inter-
polation (Enzyklopadie II A 9, p. 642 et seq.). Moreover, the formulas
that concern us here are essentially the same as those that arise in the
usual convergence proofs. It is only that the thoughts which we shall
attach to them have another shade of meaning and are designed to
adapt the material more for practical use.

I turn now to a detailed consideration of our problem, and I shall
inquire first as to the most appropriate determination of the coefficients
a, 6, ... for a given number n of terms. Bessel developed an idea here
which involves the method of least squares. The error that is made
when, for a particular x, we replace / (x) by the sum S n (x) of the first

1 Sammlung Schubert No. 32, Leipzig, 1904. See also Byerly, W. E., Fourier's
Series and Spherical Harmonics.

{Q2

Analysis: The Goniometric Functions.

2 n + 1 terms of the trigonometric series, is / (x) S n (x) , and a measure
of the closeness of representation throughout the interval ^ x ^ 2 n
(the period of / (x)) will be the sum of the squares of all the errors, that
is, the integral

The most appropriate approximation to / (x) will therefore be supplied
by that sum S n (x) for which this integral / has a minimum. It was
from this condition that Bessel determined the 2n + \ coefficients a ,
a lt . . <, a n ,b lt . . ., b n . Since we are to consider / as a function of the
2n + 1 quantities , . . ., b n , we have, as necessary conditions for a
minimum:

(2)

Since / is an essentially positive quadratic function of a Q , . . ., b n , it is
easy to see that the values of the variables determined by these 2n + 1
equations really yield a minimum.

If we differentiate under the sign of integration, the equations (2)
take the form

(20

r^n

/ [/(*) S n (*
Jo

= O l ... l f '" r [/W-S ll (^)]si
.Jo

Now the integrals of the products of S n (x) by a cosine or a sine can
be much simplified. We have, namely, for v = 0, 1 , . . . , n,

I S n (x)cosvxdx = ^ cosvxdx + a* co$xcosvxdx+"*+a n ^cosnxcosvxdx

Jo 2 JO JO JO

/2jr r2n

+ 6J smxcosvxdx+'"+b n l sinnxcosvxdx.
Jo Jo

According to known elementary integral properties of the goniometric
functions, all the terms on the right vanish, with the exception of the
cosine term with index v, which takes the value a v *n, so that

S n (x)cosvxdx = i

(v = 0, 1, . . . , n) .

Trigonometric series.

This result holds also for v = 0, by virtue of our having given to a
the factor . Similarly, we have also

/ S n (x) sinvxdx = b v n , (v = 1, . . . , n) .

From these simple relations, it follows that each of the equations (2')
contains only one of the 2^ + 1 unknowns. We can therefore write
down their solutions immediately in the form

(3)

j r2n

a v= f(x)cosvxdx, (v = 0, 1, . . . , n),

\ r 2n

b v = / f(x) sinvxdx, (v = 1, . . . , n) .

ft J o

If we make use of these values of the coefficients in S n (x) , as we
shall from now on, / actually becomes a minimum, and its value is
found to be

-f

v=l

It is important to notice that the values of the coefficients a , b
which result from our initially assumed form of S n (x) are independent
of the special number n, and that, furthermore, the coefficient belonging
to a term cosrx or sin vx has precisely the same value, whether one
uses this term alone or together with any of the others, in approximating
to / (x) according to the same principle. If we attempt, namely, to
make the best possible approximation to / (x) means of a single cosine
term a v cosvx, that is, so that

[f(x) a v cosvx]*dx = Minimum

we find for a v the same value that was deduced above. This fact makes
this method of approximation especially convenient in practice. If,
for example, one has been led to represent a function by a single multiple
of sin x, because its behaviour resembled the sine, and finds that the
approximation is not close enough, one can add on more terms, always
according to the principle of least squares, without having to alter the
term already found.

I must now show how the sums S n (x) , determined in this way,
actually tend toward the function / (x) . For such an inquiry it seems
to me desirable to proceed, in a sense, experimentally, after the method
of natural scientists, namely by first drawing for a few concrete cases
the approximating curves S n (x) . This gives a vivid picture of what
happens, and, even for persons without special mathematical gift, it
will awaken interest, and will show the need of mathematical explanation.

Klein, Elementary Mathematics. 1 3

194 Analysis: The Goniometric Functions.

In a former course of lectures (Winter semester 1903-1904) when
I discussed these things in detail, my assistant, Schimmack, made such
drawings, some of which I shall show you in the original and on the
screen.

1. We get simple and instructive examples of the desired kind if
we take curves made up of straight line segments. For example, consider
the curve y = / (x) as coinciding with y = x, from x = to x = n\2\
with y = n x , from x = nj2 to x = 3 nj2 \ with y = x 2 n from
x = 3 nj2 to x = 2 n ; and as periodically repeating itself beyond the
interval considered (0,2^). If we calculate the coefficients, we find
all the coefficients a v are zero, since / (x) is an odd function, and there
remain only the sine terms. The desired series has the form

c/^\ _ 4 / sin * sin 3* , sins* \

. ^w-^ni 31- + 5f--+ -;

In Fig. 82 the course of the first and second partial sums is sketched.
The partial sums approach the given curve y = f (x) more and more

Fig. 82.

closely in that the number of their intersections with it increase continually
It should be noticed especially that the approximating curves crowd
more and more into the corners of the curve at nj2 , 3 ^/2 , . . . , although
they themselves, as analytic functions, can have no corners.

2. Let / (x) be defined as x from x = to x = n , and as x 2 n
from x = ntox = 2n, with a gap at x = n . The curve consists, then,
of parallel straight line segments through the points x = Q,2n,4tt, ...
of the x axis. If at the points of discontinuity we insert vertical lines
joining the ends of the discontinuous segments, the function will be
represented by an unbroken line (see Fig. 83). It looks like the m strokes
which you all practiced when you were laming to write. Again the
function is odd, so that the cosine terms drop out, and the series becomes

Fig. 83 represents the sums of the first two, three, and four terms.
It is especially interesting here, also, to notice how they try to imitate

Trigonometric series.

195

the discontinuities of / (x), e.g., by going through zero at x = n with
ever increasing steepness.

3. As a last example (see Fig. 84) I shall take a curve which is equal
to Tt/2 between and n/2,
equal to between a/2 and
371/2, and finally equal to
n/2 between 3 ^/2 and
2^, and which continues
periodically beyond that.
If we again insert verti-
cal segments at the places
of discontinuity we get a
hookshaped curve. Here
also only the sine coeffi-
cients are different from
zero, since we have an
odd function, and the series
becomes Fig. 83.

S(x) = sinx + 2

sin 2x sin 3 A;

+ /}
^

sin 6 AT

sin ix

The law of the coefficients is not so simple here as it was before and
hence the successive approximating curves (Fig. 84 shows the third,
,. > _ x _^ fifth and sixth) are not so comparable gra-

phically as they were in the preceding cases.
We turn now to the question as to how
large the error is, in general, when we replace

Fig. 84.

f (x) , at a definite place, by the sum S n (x) . Up to this point we have
been concerned only with the integral of this error, taken for the entire
interval. Let us consider the integrals (3) (p. 193) f r th e coefficients
a vt b v and replace the variable of integration by I, to distinguish it

13*

Analysis: The Goniometric Functions.

from x, which we use to denote a definite point. Then we can write
our finite sum (1) as

1 /* 2
n (x) = /

+cosnxcosnl;

or, if we combine summands which are in the same column, we have

(*-)+cos2(*-{) + + cos(*-|)].

The series in the parenthesis can be summed easily, perhaps most con-
veniently by using the complex exponential function. I cannot go into
the details here, but we get, if we also use the fact that the periodicity
of the integrand enables us to integrate from n to -\-n\

. 2n + \ ... .

-*) .

To enable us to judge as to the value of this integral, let us first draw
the curves

, l 1

-*)

for the interval x n^S^lx + n of the axis. They obviously
have branches resembling a. hyperbola (see Fig. 85), and between these
branches the curve

2n + j ft _

oscillates back and forth with increasing frequency as n gets larger.
For = x it has the value r\ = (2n + l)/(2^) which increases with n.

/ + JT
7? '^1
-Jt

will represent simply the area lying between the r\ curve and the f axis
(shaded in the figure). Now anyone who has moderate feeling for con-
tinuity will see at once that if n increases sufficiently the* oscillation
areas to the right, as well as those to the left, being alternately positive
and negative, will compensate each other and that only the area of
the long narrow central arch will remain. But it is easy to see that
with increasing n this approaches the value / (x) == 1 , as it should.
And, in general, things turn out in this same way, provided / (x) does
not oscillate too strongly at x = f .

It is just such considerations, developed for more precise use, which
form the basis for Dirichlet's proof of convergence of the infinite trigono-
metric series.

Trigonometric series.

197

This proof was published 1 for the first time by Dirichlet in 1829
in volume 4 of Crelle's Journal. Later (1837) he gave a more popular
presentation 2 in the Repertorium der Physik by Dove and Moser. The
proof is given nowadays in most textbooks*, and I do not need to
dwell upon it here. But I must mention certain sufficient conditions
which the function / (x) must satisfy if it is to be represented by an
infinite trigonometric series. Again think of / (x) as given in the interval

Fig. 85-

x ^ 2 n and as periodically continued beyond. Dirichlet makes,
then, the following two assumptions which are called today simply
Dirichlet' s conditions :

a) The given function / (x) is segmentally continuous, i.e., it has in
the interval (0, 2 n) only a finite number of discontinuities, and is other-
wise continuous up to the points where it jumps.

b) The given function / (x) is segmentally monotone, i.e., one can
divide the interval (0,2^) into a finite number of sub-intervals, in
every one of which / (x) either does not increase or does not decrease.
In other words, / (x) has only a finite number of maxima and minima.
(This would exclude, for example, such a function as sin \jx, for which
x = o is a limit point of extrema.)

Dirichlet shows that, under these conditions, the infinite series re-
presents th*e function / (x) exactly for all values of x for which / (x) is

continuous. That is

limS, (*)=/(*).

n^oo

Moreover Dirichlet proves that, at a point of discontinuity, the series
converges also, but to a value which is the arithmetic mean of the two

1 Reprinted in Dirichlet, Werke, vol. 1, p. 117, Berlin, 1889.

? Vber die Darstellung ganz willkurlicher Funktionen durch Sinus- und
Kosinusreihen. Reprinted, Werke, vol. 1, p. 133 160, and Ostwalds Klassiker
No. 116, Leipzig, 1900.

* See Byerly, Fourier's Series and Spherical Harmonics.

198

Analysis: The Goniometric Functions.

values which / (x) approaches when x approaches the discontinuity Irom
the one side or the other. This fact is usually expressed in the form

Fig. 86 exhibits such discontinuities and the corresponding mean values.
These conditions of Dirichlet are sufficient, but by no means ne-
cessary, in order that / (x) may be represented by the series 5 (x) . On
the other hand, mere continuity of / (x) is not sufficient. In fact it is
possible to give examples of continuous functions where oscillations
cluster so strongly that the series S (x) diverges.

After these theoretical matters I shall now return to the practical
side of trigonometric series. For a detailed treatment of the questions
that arise here I refer you to the book by Runge which I mentioned

before (see p. 191). You will find
there a full treatment of the
question as to the numerical cal-
culation of the coefficients in the
series, i.e., the question as to
how, when a function is given,
one can rapidly evaluate the
integrals for a V9 b v in the most

ZJL

-^r

Fig. 86.

suitable way.

Special mechanical devices called harmonic analyzers have been
constructed for calculating these coefficients. This name has reference
to the relation which the development of a function / (x) into a trigono-
metric series has to acoustics. Such a development corresponds to the
separation of a given tone y = f (x) (where x is the time and y the
amplitude of the tone vibration) into "pure tones", that is, into pure
cosine and sine vibrations. In our collection we have an analyzer by
Coradi in Zurich, by means of which one can determine the coefficients
of six cosine and sine terms (v = 1 , 2, . . . , 6), i.e. twelve coefficients
in all. The coefficient <z /2 must be separately determined by a plani-
meter. Michelson and Stratton have made an apparatus with which
160 coefficients (v = 1 , 2, . . . , 80) can be determined. It is described
in Runge's book. Conversely, this apparatus can also sum a given
trigonometric series of 160 terms, i.e. calculate the function from the
given coefficients a v , b v . This problem also , of course, is of the greatest
practical importance.

The apparatus of Michelson and Stratton called attention anew to
a very interesting phenomenon, one which had been noticed earlier 1 but

1 According to Enzyklopadie vol. 2, 12 (Trigonometrische Reihen und Integrate),
p. 1048, H. Wilbraham was already familiar with the phenomenon under discussion
here and had treated it with a view to calculation.

Trigonometric series.

199

Fig. 87.

which, with the passage of decades, had, curiously enough, been forgotten.
In 1899 Gibbs again discussed it in Nature 1 , whence it is called Gibb's
phenomenon. Let me say a few words about it. The theorem of Birichlet
gives as the value of the infinite series, for a fixed value x, the expression
[/ (x + 0) + / (x 0)] . In the second example discussed above (to have
a concrete case in mind) the series gives the values at the isolated
points n, 3 n, . . . of the function pictured in Fig. 87.

Now the way in which we explained the matter of trigonometric
approximation was different from the Dirichlet procedure, where x is
kept fixed while n becomes infinite. We
thought of n as fixed, considered S n (x) with
variable x , and drew the successive approx-
imating curves 5 X (x) , S 2 (x) , S 3 (x) , . . . We
may now inquire, what happens to these
curves when n becomes infinite; or, to put
it arithmetically, what is the limit of S n (x)
when n becomes infinite, x being variable?
It is clear, intuitively, that the limit function cannot exhibit isolated
points, as before, but must be represented by a connected curve. It
would appear probable that this limit curve must consist of the con-
tinuous branches of y = f (x) , together with the vertical segments which
join / (x + 0) and / (x 0) at the points of
discontinuity, that is, in our example, the
curve would be shaped like a German m, as
is shown in Fig. 83. The fact is, however, that
the vertical part of the limit curve projects
beyond / (x + 0) and / (x 0) , by a finite
amount, so that the limit curve has the re-
markable form sketched in Fig. 88.

This little superimposed tower was noticed
in the curves which the Michelson machine drew; in other words it
was disclosed experimentally. At first it was ascribed to imperfec-
tions in the ^ apparatus, but finally Gibbs recognized it as necessary.
If D = |/ (% -f- o) / (x 0)| is, in general, the magnitude of the
jump, then the length of the extension is, according to Gibbs:

: 0.28D ^0.09#.

As to the proof of this statement, it is sufficient to give it for a single
discontinuous function, e.g., the one in our example, since all other
functions with the same spring can be obtained from it by the addition
of continuous functions. This proof is not very difficult. It results

Fig. 88.

Vol. 59 (189899), p. 200. Scientific papers II, p. 158. New York 1906.

200 Analysis: The Goniometric Functions.

immediately from consideration of the integral formula for S n (x) (see
p. 196). Furthermore, if one draws a sufficient number of the approxi-
mating curves one sees quite clearly how the Gibbs point arises.

It would lead me too far afield if I were to consider further the
many interesting niceties in the behaviour of the approximating curves.
I am glad to refer you to the full and very readable article by Fejer
in Vol. 64 (1907) of the Mathematische Annalen.

With this I shall conclude the special discussion of trigonometric
series in order to wander in a field which as to its content and its history
is closely related to them.

Excursus Concerning the General Notion of Function

We must be all the more willing, in these lectures, to discuss the
notion of function, since our school reform movement advocates giving
this important concept a prominent place in instruction.

If we follow again the historical development, we notice first that
the older authors, like Leibniz and the Bernoullis, use the function
concept only in isolated examples, such as powers, trigonometric func-
tions, and the like. A general formulation is met first in the eighteenth
century.

1. With Euler, about 1750 (to use only .round numbers), we find
two different explanations of the word function.

a) In his Introductio he defines, as a function y of x , every analytic
expression in x, i.e., every expression which is made up of powers,
logarithms, trigonometric functions, and the like ; but he does not indicate
precisely what combinations are to be admitted. Moreover, he had,
already, the familiar division into algebraic and transcendental functions.

b) At the same time, a function y (x)
(see Fig. 89) was defined for him when-
ever a curve was arbitrarily drawn
(libero manus ductu) in an x , y coordi-
nate system.

^^ 2. Lagrange, about 1800, in his
Fig. 89. Theorie des fonctions analytiquqs restricts

the notion function, in comparison witlj

Euler' s second definition, by confining it to so called analytic functions,
which are defined by a power series in x. Modern usage has retained
the words analytic functions with this same meaning, where, of course,
one must recognize that this includes only a special class of the func-
tions that really occur in analysis. Now a power series

y = P(x) = a + a l x + a 2 x*+...

defines a function primarily only within the region of its convergence,
i.e., in a certain region around x = 0. A method was soon found,

-.27

Excursus Concerning the General Notion of Function. 201

however, for extending beyond this the region of definition for the
function. If, say, x l (see Fig. 90) is within the region of convergence
of P (x) , and if P (x) is resolved into a new series

which proceeds according to powers of (x x-^ f it is possible that this

may converge in a region extending beyond the first one, and so

may define y in a larger field. A repetition of

this process may extend the field still farther.

This method of analytic continuation is well

known to any one who is familiar with com-

plex function theory.

Notice, in particular, that every coefficient Fig. 90.

in the power series P (x) , and therefore the

entire function y is determined when the behavior of the function y
along an arbitrarily small segment of the x axis is known, say in the
neighborhood of x = 0. For then the values of all the derivatives of
y are known for x = 0, and we know that

y (o) = fl , /(o) = i , /'(o) = 2*2 , . . .

Thus an analytic function, in the Lagrange sense, is determined through-
out its entire course by the shape of an arbitrarily small segment. This
property is completely opposed to the behavior of a function in the
sense of Euler's second definition. There, any part of a curve can be
continued at will.

3. The further development of the function concept is due to
J. J. Fourier, one of the numerous important mathematicians who
worked in Paris at the beginning of the nineteenth century. His chief
work is the Theorie analytique de la chaleur 1 which appeared in 1822.
Fourier made the first communication, however, concerning his theories,
to the Paris Academy in 1807- This work is the source of that far
reaching method, so much used in mathematical physics today, which
can be characterized as the resolution of all problems to the integration
of partial differential equations with initial conditions, to a so called
boundary+value problem.

Fourier treated, in particular, the problem of heat conduction which,
for a simple case, may be stated as follows. The boundary of a circular
plate is kept at a constant temperature, e.g., one part at the freezing,
the other at the boiling point (see Fig. 91). What stationary temperature
is ultimately brought about by the resulting flow of heat? Boundary
values are introduced here which can be assigned independently of each
other at different parts of the boundary. Thus Euler's second definition

1 Reprinted in Fourier, CEuvres, vol. I. Paris 1888. Translated into German
by Weinstein. Berlin 1884.

202 Analysis: The Goniometric Functions.

of function comes appropriately into the foreground, as opposed to that
of Lagrange.

This definition is retained essentially by Dirichlet in the works which
we mentioned (p. 197), except that it is translated into the language
of analysis or, to use a modern term, it is arithmetized. This is in fact,
necessary. For no matter how fine a curve be drawn, it can never
define exactly the correspondence between the values of x and y. The
stroke of the pen will always have a certain width, from which it follows
that the lengths x and y which correspond to one another can be measured
exactly only to a limited number of decimal places.

Dirichlet formulated the arithmetic content of Euler's definition in
the following way. If in any way a definite value of y is determined,
corresponding to each value of x in a given interval,
egrees ^ en y ^ ca ^ ec j a f unc tion of x. Although he announced
this very general notion of a function, nevertheless he
always thought primarily of continuous functions, or
of such as were not all too discontinuous, as was done
then quite generally. People considered complicated
.100 degrees clusterings of discontinuities as thinkable, but they
Fig. 91. hardly believed that they deserved much attention.
This standpoint finds expression when Dirichlet speaks
of the development into series of "entirely arbitrary functions" (just as
Fourier had said "fonctions entierement arbitraires) even when he
formulated very precisely his Dirichlet conditions, which must be satis-
fied by all the functions he considered.

5. We must now take account of the fact that at about this time,
say around 1830, the independent development of the theory of
functions of a complex argument began; and that in the next three
decades it became the common property of mathematicians. This
development was connected, above all, with the names Cauchy, Rie-
mann, and Weierstrass. The first two start, as you know, from the
partial differential equations which bear their names, and which must
be satisfied by the real and imaginary parts u, v of the complex function

while Weierstrass defines the function by means of a power series and
the aggregate of its analytic continuations, so that he, in a sense, follows
Lagrange.

Now it is remarkable that this passage into the complex domain
brings about an agreement and connection between the two function
concepts considered above. I shall give a brief sketch of this.

Let us put z = x + iy, and consider the power series

(1) /(*) = u + iv = c + c r z + c 2 z* + - - - ,

Excursus Concerning the General Notion of Function. 203

as converging for small \z\ so that, in the terminology of Weierstrass,
it defines an element of an analytic function. We consider its values
on a sufficiently small circle of radius r , about z = , which lies entirely
within the region of convergence (see Fig. 92), i. e., we put z = x + iy
= r (cos (p + i sin <p) in the power series, and we get

/(*) = C Q + ^^(cosy + isin<p) + C 2 r 2 (cos2<p + isin2<p)
If we separate the coefficients into real and imaginary parts:

we get as the real part of / (z)

<*0

(2)

u = u(<p) = -

The sign of the imaginary part in the c was taken negative in order

that all the signs should be positive. Thus the power series for / (z)

yields for the values, on our circle, of the real part u t thought of as a

function of the angle (p, a trigonometric series

of exactly the former sort, whose coefficients z " ane

are # , r v ot v , r v p v .

Of course, these values u will be analytic
functions of <p, in the sense of Lagrange, as long
as the circle (r) lies entirely within the region of
convergence of the power series (1). But if we
allow it to coincide with the circle of conver- Fig. 92.

gence of the series (1) which bounds its region

of convergence, then the series (1) and consequently also the series (2)
will not necessarily converge any longer. Meantime it can happen that
the series (2) continues to converge, in which case the boundary values
u (<p) cannot be analytic functions in the sense of Dirichlet.

If we proceed conversely and assign to circle (r} an arbitrary distribu-
tion of values u (<p) which satisfy only the conditions of Dirichlet, then
they can be developed into a trigonometric series of the form (2), so
that the*quantities ot , <x lf . . ., ft lf /? 2 , . . . and hence the coefficients of
*the power series (1) (to within an arbitrary additive constant (i/? )/2)
will be determined. It can be shown that this power series actually
converges within the circle (r) and that the real part of the analytic
function which it determines has the values u (q>) as boundary values
on the circle (r) , or, to be more exact, that it approaches the value u (9?)
whenever a position <p is approached for which u (<p) is continuous.

The proofs of these facts are all contained in the investigations
concerning the behavior of power series on the circle of convergence.
I cannot, of course, give them here. But these remarks may serve to

204 Analysis: The Goniometric Functions.

show how, in this way, the Fourier-Dirichlet function concept and that
of Lagrange merge into each other in that the arbitrariness in the
behaviour of the trigonometric series u (<p) on the boundary of the
circle is concentrated, for the power series, into the immediate neighbor-
hood of the center.

6. Modern science has not stopped with the formulation of these
concepts. Science never rests, even though the individual investigator
may become weary. During the last three decades mathematicians,
taking a standpoint quite different from that of Dirichjet, have siezed
upon functions having the greatest possible discontinuity, which, in
particular, do not satisfy the Dirichlet conditions. The most remarkable
types of function have been found, which contain the most disagreeable
singularities "balled into horrid lumps". It becomes a problem then to
determine how far the theorems which hold for "reasonable* ' functions
still have validity for such abnormities.

; S 7. In connection with this, there has arisen, finally, a still more far
reaching generalization of the notion of function. Up to this time, a
function was thought of as always defined at every position in the
continuum made up of all the real or complex values of x, or at least
at every position in an entire interval or region. But recently the theory
of point sets, invented by G. Cantor, has made its way more and more
to the foreground, in which the continuum of all % is only an obvious
example of a set of points. From this new standpoint functions are
being considered which are defined only for the positions x of some
arbitrary set, so that in general y is called a function of x when to every
element of a set x of things (numbers or points) there corresponds an
element of a set y.

Let me point out a difference between this newest development and
the older one. The notions considered under headings 1. to 5. have
arisen and have been developed with reference primarily to applications
in nature. We need only think of the title of Fourier's work. But the
newer investigations mentioned in 6. and 7. are the result purely of
the love of mathematical research, which has taken no account whatever
of the needs of natural phenomena, and the results have indeed found
as yet no direct application. The optimist will think, of course, 'that the
time for such application is bound to come.

We shall now put our customary question as to how much of all
this should be taken up by the schools. What should the teacher and
what should the pupils know?

In this connection I should like to say that it is not only excusable
but even desirable that the schools should always lag behind the most
recent advances of our science by a considerable space of time, certainly
several decades; that, so to speak, a certain hysteresis should take place.
But the hysteresis which actually exists at the present time is in some

Excursus Concerning the General Notion of Function. 205

respects unfortunately much greater. It embraces more than a century,
in so far as the schools, for the most part, ignore the entire development
since the time of Euler. There remains, therefore, a sufficiently large
field for the work of reform. And what we demand in the way of reform
is really quite modest, if you compare it with the present state of the
science. We desire merely that the general notion of function, according
to the one or the other of Euler 's interpretations, should permeate as
a ferment the entire mathematical instruction in the higher schools.
It should not, of course, be introduced by means of abstract definitions,
but should be transmitted to the student as a living possession, by
means of elementary examples, such as one finds in large number in
Euler. For the teacher of mathematics, however, something more than
this seems desirable, at least a knowledge of the elements of complex
function theory; and although I should not make the same demand
regarding the most recent concepts in the theory of point sets, still it
seems very desirable that among the many teachers there should always
be a small number who devote themselves to these things with the
thought of independent work.

I should like to add to these last remarks a few words concerning
the important role that has been played in this entire development by
the theory of trigonometric series. You will find extensive references
to the literature of the subject in Burkhardt's Entwickelungen nach oszil-
lierenden Funktionen (especially in chapters 2, 3> 7), that "giant report 11 ,
as his friends call it, which since 1901 has been appearing serially in
volume 10 of the Jahresbericht der deulschen Mathematikervereinigung 1 . It
combines, in more than 9000 references, an amount of pertinent literature
such as you will hardly find elsewhere.

The first to come upon the representation of general functions by
means of trigonometric series was Daniel Bernoulli, the son of John
Bernouilli. He noticed, about 1750, in his study of the acoustic problem
of vibrating strings, that the general vibration of a string could be
represented by the superposition of those sine vibrations which cor-
responded to the fundamental tone and the overtones. That involves
precisely, the development into a trigonometric series of the function
vhich represents the form of the string.

Although advances were soon made in knowledge of these series,
itill no one really believed that arbitrary functions graphically given,
:ould be represented by them. At bottom, here, there was an undefined
presentiment of considerations which have become quite clear to us
low through the theory of point sets. Perhaps one assumed, without,

1 Completed in two half volumes as Heft 2 of this volume. Leipzig 1908.

A short summary appears in the Enzyklopadie der mathematischen Wissen-

ichaften, vol. 2. Burkhardt's report goes to 1850. The development from 1850

on is sketched by Hilb and Riesz in their article in the Enzyklopadie, vol. 2, C 10.]

206 Analysis: The Goniometric Functions.

of course, being able to give precise expression to the feeling, that the
"set" of all arbitrary functions, even if discontinuities are excluded,
was greater than the "set" of all possible systems of numbers a^a^,
a 2 , . . . , b lf b 2 , . . . , which represents the totality of trigonometric series.
It is only the precise concepts of the modern theory of point sets that
have cleared this up, and have shown that that judgment was false. Let
me, at this place, elaborate somewhat this important point. It is easy
to see that the entire course of a continuous function arbitrarily defined
in a given interval, say from to 2 n t is completely known if one knows
its values at all the rational positions of that interval (see Fig. 93)-
For, since the set of these rational points is dense, we can effect an

arbitrarily close approximation for any ir-
rational position, in terms of function values
at rational ones, so that, by virtue of the
continuity of the function, the value of f(x)
is known as the limit of the function values
>x at -the approximating points. Furthermore,
Fig. 93. we know that the set of all rational numbers

is denumerable (see appendix II, p. 252), i. e.,

that they can be arranged in a series in which a definite first element
is followed by a definite second, this by a definite third, and so on.
From this it follows, however, that the assignment of the arbitrary
continuous function means nothing more than the assignment of an
appropriate denumerable set of constants the function values at the
ordered rational points. But in the same way, by means, namely, of
the denumerable series of constants # , a lt b lt a 2 , b 2 , . . ., we can assign
a definite trigonometric series, so that the doubt as to whether the
totality of continuous functions was, in the nature of things, essentially
greater than that of the series, is groundless. Similar considerations hold
for functions which are discontinuous but which satisfy the Conditions
of Dirichlet. We shall have occasion later to give detailed consideration
to these matters.

The man who abruptly brushed aside all these misgivings was
Fourier and it was just this which made him so significant in tfye history
of trigonometric series. Of course, he did not base his conclusions on
the theory of point sets, but he was the first one who had the courage
to believe in the general power of series for purposes of representation.
Fortified by this belief he set up a number of series by actual calculation,
using characteristic examples of discontinuous functions, as we did a
short time back. The proofs of convergence, as we have noted, were
first given later, by Dirichlet, who, moreover, was a pupil of Fourier.
This stand of Fourier's had a revolutionary effect. That it should be
possible to represent by series of analytic functions such arbitrary
functions as these, which obeyed in different intervals such entirely

General Considerations in Infinitesimal Calculus. 207

different laws, this was something quite new and unexpected to the
mathematicians of that time. In recognition of the disclosure of this
possibility, the name of Fourier was given to the trigonometric series
which he employed, a name which has persisted to this day. To be
sure every such personal designation implies a marked one-sidedness,
even when it is not outright injustice.

In conclusion, I must mention briefly a second accomplishment of
Fourier. He considered, namely, the limiting case of the trigonometric
series when the period of the function to be represented is allowed to
become infinite. Since a function with an infinite period is simply a
non periodic function, arbitrary along the entire % axis, this limiting
case supplies a means of representing non periodic functions. The transi-
tion is brought about by introducing a linear transformation of the
argument of the series, which effects a representation of functions with
a period / instead of 2 n, and then letting I become infinite. The series
then goes over into the so called Fourier integral

/oo

f(x) = [cp (v) cosvx + w (v) sinvx] dv ,
Jo

when <p (v) , y (v) are expressed in definite manner as integrals of the
function / (x) from oo to + oo. The new thing here is that the index v
takes continuously all values from to oo, not merely the values 0, 1,
2, . . .; and that, correspondingly, b v .

We shall now leave the elementary transcendental functions, which
have hitherto been our chief concern in our remarks on analysis, and
go over to a new concluding chapter.

III. Concerning Infinitesimal Calculus Proper

' Of course I shall assume that you all know how to differentiate and
integrate, and that you have frequently used both processes. We shall
be concerned here solely with more general questions, such as the logical
and psychological foundations, instruction, and the like.

i. General Considerations in Infinitesimal Calculus
I should like to make a general preliminary remark concerning the
range of mathematics. You can hear often from non mathematicians,
especially from philosophers, that mathematics consists exclusively in
drawing conclusions from clearly stated premises; and that, in this
process, it makes no difference what these premises signify, whether they
are true or false, provided only that they do not contradict one another.
But a person who has done productive mathematical work will talk
quite differently. In fact those persons are thinking only of the crystal-
lized form into which finished mathematical theories are finally cast.

208 Analysis: Concerning Infinitesimal Calculus Proper.

The investigator himself, however, in mathematics, as in every other
science, does not work in this rigorous deductive fashion. On the con-
trary, he makes essential use of his phantasy and proceeds inductively,
aided by heuristic expedients. One can give numerous examples of
mathematicians who have discovered theorems of the greatest importance,
which they were unable to prove. Should one, then, refuse to recognize
this as a great accomplishment and, in deference to the above definition,
insist that this is not mathematics, and that only the successors who
supply polished proofs are doing real mathematics? After all, it is an
arbitrary thing how the word is to be used, but no judgment of value
can deny that the inductive work of the person who first announces
the theorem is at least as valuable as the deductive work of the one who
first proves it. For both are equally necessary, and the discovery is
the presupposition of the later conclusion.

It is precisely in the discovery and in the development of the
infinitesimal calculus that this inductive process, built up without
compelling logical steps, played such a great role; and the effective
heuristic aid was very often sense perception. And I mean here im-
mediate sense perception, with all its inexactness, for which a curve
is a stroke of definite width, rather than an abstract perception which
postulates a completed passage to the limit, yielding a one dimen-
sional line. I should like to corroborate this statement by outlining
to you how the ideas of the infinitesimal calculus were developed
historically.

If we take up first the notion of an integral, we notice that it begins
historically with the problem of measuring areas and volumes (quadra-
ture and cubature). The abstract logical definition determines the

/b
f(x) dx, i.e., the area bounded by the curve y = /(#), the %
i

axis, and the ordinates x = a , % = b , as the limit of the sum of narrow
rectangles inscribed in this area when their number increases and their
width decreases without bound. Sense perception, however, makes it
natural to define this area, not as this exact limit, but simply as the
sum of a large number of quite narrow rectangles. In fact, the necessary
inexactness of the drawing would inevitably set bounds to the further
narrowing of the rectangles (see Fig. 94).

This naive method characterizes, in fact, the thinking of the greatest
investigators in the early period of infinitesimal calculus. Let me men-
tion, first of all, Kepler who in his Nova stereometria doliorum vinario-
rum 1 was concerned with the volumes of bodies. His chief interest
here was in the measuring of casks, and in determining their most suit-
able shape. He took precisely the naive standpoint indicated above.

1 Linz on the Danube, 1615. German in Ostwalds Klassikern, No. 165- Leipzig,
1908.

General Considerations in Infinitesimal Calculus.

209

He thought of the volume of the cask, as of every other body (see Fig. 95),
as made up of numerous thin sheets suitably ranged in layers, and
considered it as the sum of the volumes of these leaves, each of which
was a cylinder. In a similar way he calculated the simple geometric
bodies, e. g., the sphere. He thought of this as made up of a great
many small pyramids with common vertex at the center (see Fig. 96).
Then its volume, according to the well known formula for the pyramid,
would be 7/3 times the sum of the bases of all the small pyramids. By
writing for the sum of these little facets simply the surface of the sphere,
or 4 n r*, he obtained 4 n r*/') , the correct formula for the volume.

Fig. 94.

Fig. 95-

Fig. 96.

Moreover, Kepler emphasizes explicitly the practical heuristic value of
such considerations, and refers, so far as rigorous mathematical proofs
are concerned, to the so called method of exhaustion. This method, which
had been used by Archimedes, determines, for example, the area of the
circle by following carefully the approximations to the area by means
of inscribed and circumscribed polygons with an increasing number of
sides. The essential difference between it and the modern method lies
in the fact that it tacitly assumes, as self evident, the existence of a
number which measures the area of the circle, whereas the modern
infinitesimal calculus declines to accept this intuitive evidence, but has
recourse to the abstract notion of limit and defines this number as the
limit of the numbers that measure the areas of the inscribed polygons.
Granted, however, the existence of this number, the method of ex-
haustion is an exact process for approximating to areas by means of
the known areas of rectilinear figures, one which satisfies rigorous
modern demands. The method is, however, very tedious in many cases,
and ill suited to the discovery of areas and volumes. One of Archimedes
writings 1 , discovered by H. Heiberg in 1906, shows, in fact, that he did
not use the method of exhaustion at all in his investigations. After
he had first obtained his results by some other method, he developed
the proof by exhaustion in order to meet the demands of that time as
to rigor. For the discovery of his theorems he used a method which
included considerations of the center of gravity and the law of the lever,
and also of intuition, such as, for example, that triangles and parabolic

1 Already referred to on p. 80.
Klein, Elementary Mathematics.

210

Analysis: Concerning Infinitesimal Calculus Proper.

segments consist of series of parallel chords, or that cylinders, spheres,
and cones are made up of series of parallel circular discs.

Returning now to the seventeenth century, we find considerations
analogous to those of Kepler in the book of the Jesuit Bonaventura
Cavalieri: Geometria indivisibilibus continuorum nova quadam ratione
promota 1 where he sets up the principle called today by his name: Two
bodies have equal volumes if plane sections equidistant from their bases
have equal areas. This principle of Cavalieri is, as you know, much used
in our schools. It is believed there that integral calculus can be avoided
in this way, whereas this principle belongs, in fact, entirely to the
calculus. Its establishment by Cavalieri amounts precisely to this, that
he thinks of both solids as built up of layers of thin leaves which, ac-
cording to the hypothesis, are congruent in pairs, i.e., one of the bodies
could be transformed into the other by translating its individual leaves
(see Fig. 97) ; but this could not alter the volume, since this consists of
the same summands before and after the translation.

Naive sense perception leads in the same
way to the differential quotient of a function,
i. e., to the tangent to the curve. In this case,
we can replace (and this is the way it was
actually done) the curve by a polygonal line
(see Fig. 98) which has on the curve a suffi-
cient number of points, as vertices, taken close
together. From the nature of our sense percep-
tion we can hardly distinguish the curve from this aggregate of points
and still less from the polygonal line. The tangent is now defined
outright as the line joining two successive points, that is, as the
prolongation of one of the sides of the polygon.
From the abstract logical standpoint, this line
remains only a secant, no matter how close
together the points are taken; and the tangent
is only the limiting position approached by the
secant when the distance between the points
approaches zero. Again, from this naive stand-
point, the circle of curvature is thought of as the circle which passes
through three successive polygon vertices, whereas exact procedure
defines the circle of curvature as the limiting position of this circle
when the three points approach each other.

The force of conviction inherent in such naive guiding reflections is,
of course, different for different individuals. Many and I include
myself here find them very satisfying. Others, again, who are gifted
only on the purely logical side, find them thoroughly meaningless and
are unable to see how anyone can consider them as a basis for mathe-

1 Bononiae, 1635. First edition, 1653.

Fig. 97.

Fig, 98.

General Considerations in Infinitesimal Calculus.

rnatical thought. Yet considerations of this sort have often formed
the beginnings of new and fruitful speculations.

Moreover, these naive methods always rise to unconscious importance
whenever in mathematical physic, mechanics, or differential geometry
a preliminary theorem is to be set up. You all know that they are
very serviceable then. To be sure, the pure mathematician is not
sparing of his scorn on these occasions. When I was a student it was
said that the differential, for a physicist, was a piece of brass which he
treated as he did the rest of his apparatus.

In this connection, I should like to commend the Leibniz notation,
the leading one today, because it combines with a suitable suggestion
of nai've intuition, a certain reference to the abstract limit process which
is implicit in the concept. Thus, the Leibniz symbol dy/dx, for the
differential quotient, reminds one, first that it comes from a quotient;
but the d, as opposed to the A which is the usual symbol for finite
difference, indicates that something new has been added, namely, the
passage to the limit. In the same way, the integral symbol / y dx sug-
gests the origin of the integral from a sum of small quantities. However,
one does not use the usual sign 2 for a sum, but rather a conventionalized
5*, which indicates here that something new has entered the process
of summation.

We shall now discuss with some detail the logical foundation of
differential and integral claculus, and at the same time consider it in
its historical development.

1. The principal idea, as the subject is taught, in general, at the
university (I need only briefly to refresh your memory here) is that
infinitesimal calculus is only an application of the general notion of limit.
The differential quotient is defined as the limit of the quotient of
corresponding finite increments of variable and function

dy ,. Ay
~ = lim ~-

dx AX^ AX

provided that this limit exists; and not at all as a quotient in which dy
,and dx have an independent meaning. In the same way, the integral
is defined as the limit of a sum:

/b
ydx = lim
-i Axt=Q

where the Axi are finite parts of the interval a^x^b, the % cor-
responding arbitrary values of the function in that interval, and all
iheAxi are to converge toward zero; but y dx does not have any actual
significance as, say, a summand of a sum. These designations are
retained for the reasons of expediency which we mentioned above.

* It is remarkable that many are unaware that f has this meaning.

14*

212 Analysis: Concerning Infinitesimal Calculus Proper.

2. The conception as we have thus characterized it is set forth in
precise form by Newton himself. I refer you to a place in his principal
work, the Philosophiae Naturalis Principia Mathematical of 1687-' "Ulti-
mae rationes illae, quibuscum quantitates evanescunt, revera non sunt
rationes quantitatum ultimarum, sed limites, ad quos quantitatum sine
limite descrescentium rationes semper appropinquant, et quos propius
assequi possunt, quam pro data quavis differentia, nunquam vero trans-
gredi neque prius attingere quam quantitates diminuuntur in infinitum."
Moreover, Newton avoids the infinitesimal calculus, as such, in the
discussions in this work, although he certainly had used it in deriving
his results. For, the fundamental work in which he developed his method
of infinitesimal calculus was written in 1671, although it did not appear
until 1736. It bears the title Methodus Fluxionum et Serierum Infini-
tarum*.

In this, Newton develops the new calculus in numerous examples,
without going into fundamental explanations. He makes connection
here with a phenomenon of daily life which suggests a passage to a
limit. If one considers, namely, a motion x = f (t) on the x axis in the
time t, then every one has a notion as to what is meant by the velocity
of this motion. If we analyze this motion it turns out that we mean
the limiting value of the difference quotient Ax/ At. Newton made this
velocity of x with respect to the time the basis of his developments. He
called it the "fluxion" of x and wrote it #. He considered all the variables
x, y as dependent on this fundamental variable t, the time. Accordingly
the differential quotient dy/dx appears as the quotient of two fluxions
y/x which we now should write more fully (dy/dt: dxjdt).

3. These ideas of Newton were accepted and developed by a long
series of mathematicians of the eighteenth century, who built up the
infinitesimal calculus, with more or less precision, upon the notion of
limit. I shall select only a few names: C. Maclaurin, in his Treatise of
Fluxions*, which as a textbook certainly had a wide influence; then
d'Alembert, in the great French Encyclopedie Methodique; and finally
Kastner 4 , in Gottingen, in his lectures and books. Euler belongs pri-
marily in this group although, with him, other tendencies also came
to the front.

4. It was necessary to fill out an essential gap in all these develop-
ments, before one could speak of a consistent system of infinitesimal
calculus. To be sure, the differential quotient was defined as a limit,
but there was lacking a method for estimating, from it, the increment

1 New edition by W. Thomson and H. Blackburn, Glasgow, 1871, p. 38i

2 Newtoni, J., Opuscula Mathematica, philosophica, et philologica, vol.1, p. 29.
Lausanne, 1744.

a Edinburgh, 1742.

4 Kastner, A. G., Anfangsgrunde der Analysis des Unendlichen, Gottingen, 1760.

General Considerations in Infinitesimal Calculus.

213

of the function in a finite interval. This was supplied by the mean value
theorem] and it was Cauchy's great service to have recognized its funda-
mental importance and to have made it the starting point accordingly
of differential calculus. And it is not saying too much if, because of
this, we adjudge Cauchy as the founder of exact infinitesimal calculus
in the modern sense. The fundamental work in this connection, based
on his Paris lectures, is his Resume des Lemons sur le Calcul Infinitesimal 1 ,
together with its second edition, of which only the first part, Lefons sur
le Calcul Differentiel*, was published.

The mean-value theorem, as you know, may be stated as follows. //
a continuous function f (x) possesses a differential quotient f(x) every-
where in a given interval, then there must be a point x + /M between x
and % + h such that

f(x + h)= f(x) + h*f(x + i)h) , (0 < * <1).

Note here the appearance of that ft, peculiar to the mean value theorems,
and which to beginners often seems so strange at first. Geometrically.

jc+h.

x+h

Fig. 99.

Fig. 100.

the theorem is fairly obvious. It says, merely, that between the points
x and x + h on the curve there is a point x + fth on the curve at
which the tangent is parallel to the secant joining the points x and
x + h (see Fig. 99).

5. How can one give an exact arithmetic proof of the mean value
theorem, without appealing to geometric intuition? Such a proof could
only mean, of course, throwing the theorem back upon arithmetic de-
finitions of variable, function, continuity etc., which would have to be
set up in* advance in abstract, precise form. For this reason such a
rigorous proof had to wait for Weierstrass and his followers, to whom,
also, we owe the spread of the modern arithmetic concept of the number
continuum. I shall try to give you the characteristic points of the
argument.

In the first place, it is easy to make this theorem depend on the
case where the secant is horizontal, i.e. / (x) = f (x + h) (see Fig. 100).
One must then prove the existence of a place where the tangent is

1 Paris, 1823- OEuvres completes, 2nd series, vol. 4, Paris, 1899-

2 Paris, 1829. CEuvres completes, 2nd series, vol. 4, Paris, 1899-

214 Analysis: Concerning Infinitesimal Calculus Proper.

horizontal. To do this we can use the theorem of Weierstrass that
every function which is constinubus throughout a closed interval actually
reaches a maximum, and also a minimum value, at least once in that
interval. Because of our assumption, one of these extreme values of
our function must lie within the interval (x, x + h), provided we ex-
clude the trivial case in which / (x) is a constant. Let us suppose that
there is a maximum (the case of a minimum is treated in the same
way) and that it occurs at the place x + &h. It follows that / (x)
cannot have larger values, either to the right or to the left, i.e., the
difference quotient to the right is negative, or zero, and to the left,
positive or zero. Since the differential quotient exists, by hypothesis,
at every place in the interval, its value at x + till can be looked upon
as the limit of values which are either not positive or not negative,
according as one thinks of it as a progressive or a regressive derivative.
Therefore it must have the value zero, the tangent at x = $h is hori-
zontal, and the theorem is proved.

The scientific mathematics of today is built upon the series of
developments which we have been outlining. But an essentially different
conception of infinitesimal calculus has been running parallel with this
through the centuries.

1. What precedes harks back to old metaphysical speculations con-
cerning the structure of the continuum according to which this was
made up of ultimate indivisible infinitely small parts. There were already,
in ancient times, suggestions of these indivisibles and they were widely
cultivated by the scholastics and still further by the Jesuit philosophers.
As a characteristic example I recall the title of Cavalieri's book, men-
tioned on p. 210 Geometria Indivisibilibus Continuomm Promota, which
indicates its true nature. As a matter of fact, he considers intuitive
mathematical approximation in a secondary way only. He actually
considers space as consisting of ultimate indivisible parts, the "indivisi-
bilia". In this connection it would be interesting and important to
know the various analyses to which the notion of the continuum has
been subjected in the course of centuries (arid milleniums).

2. Leibniz, who shares with Newton the distinction of having in-
vented the infinitesimal calculus, also made use of such ideas. The
primary thing for him was not the differential quotient thought of as
a limit. The differential dx of the variable x had for him actual existence
as an ultimate indivisible part of the axis of abscissas, as a quantity
smaller than any finite quantity and still not zero ("actually*' infinitely
small). In the same way the differentials of higher order d*x, d*x, . . .
are defined as infinitely small quantities of second, third, . . . order,
each of which is "infinitely small in comparison with the preceding".
Thus one had a series of systems of qualitatively different magnitudes,
According to the theory of indivisibles, the area bounded by the curve

General Considerations in Infinitesimal Calculus, 215

y = y (x) and the axis of abscissas is the direct sum of all the individual
ordinates. It is because of this view that Leibniz, in his first manuscript
on integral calculus (1675), writes jy and not fydx.

This point of view, however, is by no means the only one which
interested Leibniz. Sometimes he uses the notion of mathematical
approximation, where, for example, the differential dx is a finite segment
but so small that, for that interval, the curve is not appreciably different
from the tangent. The above metaphysical speculations are surely only
idealizations of these simple psychological facts.

But there is a third direction for the mathematical ideas of Leibniz,
one which is especially characteristic of him. It is his formal point of
view. I have frequently reminded you that we can look upon Leibniz
as the founder of formal mathematics. His thought here is as follows.
It makes no difference what meaning we attach to the differentials,
or whether we attach any meaning whatever to them. If we define
appropriate rules of operation for them, and if we employ these rules
properly, it is certain that something reasonable and correct will result.
Leibniz refers repeatedly to the analogy with complex numbers, con-
cerning which he had corresponding notions. As to these rules of ope-
ration for differentials he was concerned chiefly with the formula

The mean value theorem shows that this is correct only if one writes
/' (x + & dx) instead of /' (x) ; but the error which one commits by
writing /'(#) outright is infinitely small, of higher (second) order, and
such quantities are to be neglected (this is the most important formal
rule) in operations with differentials.

The chief publications of Leibniz are contained in that famous first
scientific journal, the Ada Eruditorum 1 ', in the years 1684, 1685, and
1712. In the first volume, you find, under the title Nova methodus pro
maximis et minimis (p. 467 et seq.), the very first publication concerning
differential calculus. In this Leibniz merely develops the rules for
differentiation. The later works give also expositions of principles, where
preference, is given to the formal standpoint. In this connection, the
sjiort article of the year 1712 2 , one of the last years of his life, was
especially characteristic. In this he speaks outright of theorems and
definitions which are only "tolemnter vera" or French "passables" :
"Rigorem quidem non sustinent, habent tamen usum magnum in calcu-
lando et ad artem inveniendi universalesque conceptus valent." He
has reference here to complex numbers as well as to the infinite. If

1 Translated, in part, in Ostwalds Klassikern No. 1 62. Edited by G. Kowalewski,
Leipzig, 1908. Also in Leibniz, Mathematische Schriften. Edited by K. J. Ger-
hardt, from 1849 on.
, a Observatio . . .; et de vero sensu methodi infinitesimalis, p. 167 169-

21 6 Analysis: Concerning Infinitesimal Calculus Proper.

we speak, perhaps, of the infinitely small, then "commoditati expressio-
nis seu breviloquio mentalis inservimus, sed non nisi toleranter vera
loquimur, quae explicatione rigidantur."

3. From Leibniz as center the new calculus spread rapidly over the
continent and we find each of his three points of view represented.
I must mention here the first textbook of differential calculus that ever
appeared, the Analyse des Infiniment Petits pour V Intelligence des
Courbes 1 by Marquis de T Hospital, a pupil of Johann Bernoulli, who
for his part, had absorbed the new ideas from Leibniz with surprising
speed and had himself published the first textbook on the integral
calculus 2 . Both books represent the point of view of mathematics of
approximation. For example, a curve is thought of as a polygon with
short sides, a tangent as the prolongation of one of these sides. In
Germany, the differential calculus according to Leibniz was spread
widely by Christian Wolff, of Halle, who published the contents of his
lectures in Elementa matheseos universal*. He introduces the differentials
of Leibniz immediately, at the beginning of the differential calculus,
although he emphasizes particularly that they have no actual equivalent
of any kind. And, indeed, as an aid to our intuition he develops his
views concerning the infinitely small in a manner which savors thoroughly
of mathematics of approximation. Thus he says, by way of example,
that for purposes of practical measurement, the height of a mountain
is not noticeably changed by adding or removing a particle of dust.

4. You will also frequently find the metaphysical view which ascribes
an actual existence to the differentials. It has always had supporters,
especially on the philosophic side, but also among mathematical physi-
cists. One of the most prominent here is Poisson, who, in the preface
to his celebrated Traite de Mecanique*, expressed himself strongly to
the effect that the infinitely small magnitudes are not merely an aid
in investigation but that they have a thoroughly real existence.

5. Due probably to the philosophic tradition, this concept went
over into textbook literature and plays a marked role there even today.
As an example, I mention the textbook by Liibsen Einleitung in die
Infinitesimalrechnung* which appeared first in 1855 and which had for
a long time an extraordinary influence among a large part of the public.
Everyone, in my day, certainly had Lubsen's book in his hand, either
when he was a pupil, or later, and many received from it the first

1 Paris, 1696; second edition, 1715-

[ 2 Translated in Ostwalds Klassikern No. 194. Edited by G. Kowalewski.
Job. Bernoulli's Differentialrechnung was discovered and discussed a short time
ago by P. Schafheitlin. Verhandlungen der Naturforscher-Gesellschaft in Basel,
vol. 32 (1921).]

3 Appeared first in 1710. Editio nova Hallae, Magdeburgiae, 1742, p. 545.

4 Part I, second edition, p. 14. Paris, 1833-

5 Eighth edition, Leipzig, 1899-

General Considerations in Infinitesimal Calculus. 217

stimulation to further mathematical study. Liibsen defined the diffe-
rential quotient first by means of the limit notion; but along side of
this he placed (after the second edition) what he considered to be the
true infinitesimal calculus a mystical scheme of operating with infinitely
small quantities. These chapters are marked with an asterisk to indicate
that they bring nothing new in the way of result. The differentials are
introduced as ultimate parts which arise, for example, by continued
.halving of a finite quantity an infinite, non assignable number of times ;
and each of these parts "although different from absolute zero is never-
theless not assignable, but an infinitesimal magnitude, a breath, an
instant". And then follows an English quotation: "An infinitesimal is
the spirit of a departed quantity" (p. 59, 60). Then in another place
(p. 76): "The infinitesimal method is, as you see, very subtle, but
correct. If this is not manifest from what has preceded, together with
what follows, it is the fault only of inadequate exposition." It is cer-
tainly very interesting to read these passages.

As companion piece to this I put before you the sixth edition of
the widely used Lehrbuch der Experimentalphysik by Wiillner 1 . The
first volume contains a brief preliminary exposition of infinitesimal
calculus for the benefit of those students of natural science or medicine
who have not acquired, at the gymnasium, that knowledge of calculus
which is indispensable for physics. Wiillner begins (p. 31) with the
explanation of the meaning of the infinitely small quantity dx t then
follows with the explanation for the second differential d z x, which, of
course, is more difficult. I urge you to read this introduction with the
eye of the mathematician and to reflect upon the absurdity of sup-
pressing infinitesimal calculus in the schools because it is too difficult,
and then of expecting a student in his first semester to gain an under-
standing of it from this ten page presentation, which is not only far from
satisfying, but very hard to read!

The reason why such reflections could so long hold their place
abreast of the mathematically rigorous method of limits, must be sought
probably in the widely felt need of penetrating beyond the abstract
ogical formulation of the method of limits to the intrinsic nature of
xmtinuous magnitudes, and of forming more definite images of them
:han were supplied by emphasis solely upon the psychological moment
tfhich determined the concept of limit. There is one formulation which
is characteristic, which is due, I believe, to the philosopher Hegel, and
which formerly was frequently used in textbooks and lectures. It
declares that the function y = / (x) represents the being, the differential
quotient dy[dx, however, the becoming, of things. There is assuredly
something impressive in this, but one must recognize clearly that such

1 Leipzig, 1907.

218 Analysis: Concerning Infinitesimal Calculus Proper.

words do not promote further mathematical development because this
must be based upon precise concepts.

In the most recent mathematics, "actually" infinitely small quantities
have come to the front again, but in entirely different connection,
namely in the geometric investigations of Veronese and also in Hilbert's
Grundlagen der Geometric 1 * The guiding thought of these investigations
can be stated briefly as follows: A geometry is considered in which
x = a (a an ordinary real number) determines not only one point on.
the x axis, but infinitely many points, whose abscissas differ by finite
multiples of infinitely small quantities of different orders TJ, f, . . .
A point is thus determined only when one assigns

x = a + by + c + ,

where a, 6, c, . . . are ordinary real numbers, and the 17, , . . . actually
infinitely small quantities of decreasing orders. Hilbert uses this guiding
idea by subjecting these new quantities i] , ?, . . . to such axiomatic
assumptions as will make it evident that one can operate with them
consistently. To this end it is of chief importance to determine appro-
priately the relation as to. size between x and a second quantity x l = a t
+ bitf + Cif + . The first assumption is that x > or < x l if
a > or < ! ; but if a = a l , the determination as to size rests with the
second coefficient, so that x^x l according as b ^ b^\ and if, in addition,
b &! , the decision lies with the c , etc. These assumptions will be
clearer to you if you refrain from attempting to associate with the
letters any sort of concrete representation.

Now it turns out that, after imposing upon these new quantities
this rule, together with certain others, it is possible to operate with
them as with finite numbers. One essential theorem, however, which
holds in the system of ordinary real numbers, now loses its validity,
namely the theorem : Given two positive numbers e , a, it is always possible
to find a finite integer n such that n e> a, no matter how small e is
nor how large a may be. In fact, it follows immediately from the above
definition that an arbitrary finite multiple n 17 of r) is smaller than
any positive finite number a, and it is precisely this property that
characterizes the ij as an infinitely small quantity. In the same way
n < YI , that is, is an infinitely small quantity of higher order than r\ .

This number system is called non-Archimedean. The above theorem
concerning finite numbers is called, namely, the axiom of Archimedes,
because he emphasized it as an unprovable assumption, or as a funda-
mental one which did not need proof, in connection with the numbers
which he used. The denial of this axiom characterizes the possibility
of actually infinitely small quantities. The name Archimedean axiom,
however, like most personal designations, is historically inexact. Euclid

1 Fifth edition, Leipzig, 1922.

/General Considerations in Infinitesimal Calculus. 219

gave prominence to this axiom more than half a century before Archi-
medes ; and it is said not to have been invented by Euclid, either, but,
like so many of his theorems, to have been taken over from Eudoxus
of Knidos. The study of non-Archimedean quantities 1 , which have
been used especially as coordinates in setting up a non-Archimedean
geometry, aims at deeper knowledge of the nature of continuity and
belongs to the large group of investigations concerning the logical de-
pendence of different axioms of ordinary geometry and arithmetic. For
this purpose, the method is always to set up artificial number systems
for which only a part of the axioms hold, and to infer the logical in-
dependence of the remaining axioms from these.

The question naturally arises whether, starting from such number
systems, it would be possible to modify the traditional foundations of
infinitesimal calculus, so as to include actually infinitely small quantities
in a way that would satisfy modern demands as to rigor; in other words,
to construct a non-Archimedean analysis. The first and chief problem
of this analysis would be to prove the mean-value theorem

from the assumed axioms. I will not say that progress in this direction
is impossible, but it is true that none of the investigators who have
busied themselves with actually infinitely small quantities have achieved
anything positive. . .

I remark for your orientation that, sincy Cauchy's time, the words
infinitely small are used in modern textbooks in a somewhat changed
sense. One never says, namely, that a quantity is infinitely small, but
rather that it becomes infinitely small; which is only a convenient ex-
pression implying that the quantity decreases without bound toward zero.

We must bear in mind the reaction which was evoked by the use
of infinitely small quantities in infinitesimal calculus. People soon
sensed the mystical, the unproven, in these ideas, and there arose often
a prejudice, as though the differential calculus were a particular philo-
sophical system which could not be proved, which could only be believed
or, to put it bluntly, a fraud. One of the keenest critics, in this sense,
^was the philosopher Bishop Berkeley, who in the little book The Analyst*
assailed in an amusing manner the lack of clearness which prevailed
in the mathematics of his time. Claiming the privilege of exercising the
same freedom in criticizing the principles and methods of mathematics
"which the mathematicians employed with respect to the mysteries of
religion", he launched a violent attack upon all the methods of the new

t 1 The so-called horn-shaped angles, known already to Euclid, are examples
of non- Archimedean quantities. Compare also the excursus, in the second volume
of this work, in connection with the critique of Euclid's Elements.]
2 London, 1734.

220 Analysis: Concerning Infinitesimal Calculus Proper.

analysis, the calculus with fluxions as well as the operation with diffe-
rentials. He came to the conclusion that the entire structure of analysis
was obscure and thoroughly unintelligible.

Similar views have often maintained themselves even up to the
present time, especially on the philosophical side. This is due, perhaps,
to the fact that acquaintance here is confined to the operation with
differentials; the rigorous method of limits, a rather recent development,
has not been comprehended. As an example, let me quote from Bau-
mann's Raum, Zeit und Mathematik 1 which appeared in the sixties:
"Thus we discard the logical and metaphysical justification, which
Leibniz gave to calculus, but we decline to touch this calculus itself.
We look upon it as an ingenious invention which has turned out well
in practice; as an art rather than a science. It cannot be constructed
logically. It does not follow from the elements of ordinary mathe-
matics . . ."

This reaction against differentials accounts also for the attempt by
Lagrange, already mentioned, in his Theorie des Fonctions Analytiques,
published in 1 797, to eliminate from the theory not only infinitely small
quantities, but also every passage to the limit. He confined himself,
namely, to those functions which are defined by power series

and he defines formally the "derived function /' ' (x)" (he avoids charac-
teristically the expression differential quotient and the sign dy/dx) by
means of a new power series

Consequently he talks of derivative calculus instead of differential calculus.
This presentation, of course, could not be permanently satisfactory.
In the first place, the concept of function used here is, as we have
shown, much too limited. More than that, however, such thoroughly
formal definitions make a deeper comprehension of the nature of the
differential coefficient impossible, and take no account of what we called
the psychological moment they leave entirely unexplained just f why one
should be interested in a series obtained in such a peculiar way. Finally , t
one can get along without giving any thought to a limit process only
by disregarding entirely the convergence of these series and the question
within what limits of error they can be replaced by finite sums. As soon
as one begins a consideration of these problems, which is essential, of
course, for any actual use of the series, it is necessary to have recourse
precisely to that notion of limit, the avoidance of which was the purpose
of inventing the system.

1 Vol. 2, p 55- Berlin, 1869.

General Considerations in Infinitesimal Calculus. 221

It would be fitting, perhaps, to say a few words about the differences
of opinion concerning the foundations of calculus, as these come up,
even today, beyond the narrow circle of professional mathematicians.
I believe that we can often find here the preliminary conditions for
understanding, in considerations very similar to those which we set forth
respecting the foundations of arithmetic (p. 13). In every branch of
mathematical knowledge one must separate sharply the question as to
the inner logical consistency of its structure from that as to the justi-
fication for applying its axiomatically and (so to speak) arbitrarily
formulated notions and theorems to objects of our external or internal
perception. George Cantor 1 makes the distinction, with reference to
whole numbers, between immanent reality, which belongs to them by
virtue of their logical definability, and transient reality, which they
possess by virtue of their applicability to concrete things. In the case
of infinitesimal calculus, the first problem is completely solved by means
of those theories which the science of mathematics has developed in
logically complete manner (through the use of the concept of limit).
The second question belongs entirely to the theory of knowledge, and
the mathematician contributes only to its precise formulation when he
separates from it and solves the first part. No pure mathematical work
can, from its very nature, supply any immediate contribution to its
solution. (See the analogous remarks on arithmetic, p. 13 et seq.) All
disputes concerning the foundations of infinitesimal calculus labor under
the disadvantage that these two entirely different phases of the problem
have not been sharply enough separated. In fact, the first, the purely
mathematical part, is established here precisely as in all other branches
of mathematics, and the difficulties lie in the second, the philosophical
part. The value of investigations which press forward in this second
direction takes on especial importance in view of these considerations;
but it becomes imperative to make them depend upon exact knowledge
of the results of the purely mathematical work upon the first problem.

I must conclude with this excursus our short historical sketch of
the development of infinitesimal calculus. In it I was obliged of course
to confine myself to an emphasis of the most important guiding notions.
It shoulcl be extended, naturally, by a thorough-going study of the
entire literature of that period. You will find many interesting references
in the lecture given by Max Simon at the Frankfurt meeting of the
natural scientists of 1896: Zur Geschichte und Philosophie der Differential-
rechnung.

If we now examine, finally, the attitude towards infinitesimal
calculus in school instruction, we shall see that the course of its historical
development is mirrored there to a certain extent. In earlier years,

1 Mathematische Annalen, vol. 21 (1883), p. 562.

222: Analysis: Concerning Infinitesimal Calculus Proper.

where infinitesimal calculus was taught in the schools, there was by no
means a clear notion of its exact scientific structure as based on the
method of limits. 'At least this was manifest in the textbooks, and it
was doubtless the same in the schools. This method cropped up in a
vague way at most, whereas operations with infinitely small quantities
and sometimes also derivative calculus, in the sense of Lagrange, came
to the front. Such instruction, of course, lacked not only rigor but
intelligibility as well, and it is easy to see why a marked aversion arose
to the treatment of infinitesimal calculus at all in the schools. This
culminated in the seventies and eighties in an official order forbidding
this instruction even in the "real" institutions.

To be sure this did not entirely prevent (as I indicated earlier) the
using of the method of limits in the schools, where it was necessary one
merely avoided that name, or one even thought sometimes that some-
thing else was being taught. I shall mention here only three examples
which most of you will recall from your school days.

a) The well known calculation of the perimeter and the area of the
circle by an approximation which uses the inscribed and circumscribed
regular polygons is obviously nothing but an integration. It was em-
ployed, even in ancient times, and was used particularly by Archimedes ;
in fact, it is owing to its classical antiquity that is has been retained
in the schools.

b) Instruction in physics, and particularly in mechanics, necessarily
involves the notions of velocity and acceleration, and their use in various
deductions, including the laws of falling bodies. But the derivation of
these laws is essentially identical with the integration of the differential
equation z" = g by means of the function z = \ gt 2 + at + b, where
a, b are constants of integration. The schools must solve this problem,
under pressure of the demands of physics, and the means which they
employ are more or less exact methods of integration, of course disguised.

c) In many North German schools the theory of maxima and minima
was taught according to a method which bore the name of Schellbach,
the prominent mathematical pedagogue of whom you all must have
heard. According to this method one puts

in order to obtain the extremes of the function y = / (x) , But that is
precisely the method of differential calculus, only that the word differen-
tial quotient is not used. It is certain that Schellbach used the above
expression only because differential calculus was prohibited in the
schools and he nevertheless did not want to miss these important
notions. His pupils, however, took it over unchanged, called it by his
name, and so it came about that methods which Fermat, Leibniz, and

Taylor's Theorem.

Newton had possessed were put before the pupils under the name of
Schellbach!

Let me now indicate, finally, the attitude toward these things of
our reform tendency, which is now gaining ground more and more in
Germany, as well as elsewhere, especially in France, and which we hope
will control the mathematical instruction of the next decades. We
desire that the concepts which are expressed by the symbols y = f (x) ,
dy/dx, fydx be made familiar to pupils , under these designations; not,
indeed, as a new abstract discipline, but as an organic part of the total
instruction; and that one advance slowly, beginning with the simplest
examples. Thus one might begin, with pupils of the age of fourteen
and fifteen, by treating fully the functions y = ax + b (a,b definite
numbers) and y = x 2 , drawing them on cross-section paper, and letting
the concepts slope and area develop slowly. But one should hold to
concrete examples. During the next three years this knowledge could
be gathered together and treated as a whole, the result being that the
pupils would come into complete possession of the beginnings of in-
finitesimal calculus. It is essential here to make it clear to the pupil
that he is dealing, not with something mystical, but with simple things
that anyone can understand.

The urgent necessity of such reforms lies in the fact that they are
concerned with those mathematical notions which govern completely
the applications of mathematics which are being made today in every
possible field, and without which all studies at the university, even the
simplest studies in experimental physics, are suspended in mid air.
I can be content with these few hints, chiefly because this subject is
fully discussed in Klein-Schimmack (referred to on p. 3).

In order to supplement these general considerations with something
which again is concrete I shall now discuss in some detail an especially
important subject in infinitesimal calculus.

2. Taylor's Theorem

I shall proceed here in a manner analogous to the plan I followed
with trigonometric series. I shall depart, namely, from the usual
.treatment in the textbooks by bringing to the foreground the finite
series, so important in practice, and by aiding the intuitive grasp of
the situation by means of graphs. In this way it will all seem elementary
and easily comprehensible.

We begin with the question whether we can make a suitable appro-
ximation to an arbitrary curve y = / (x) , for a short distance, by means
of curves of the simplest kind. The most obvious thing is to replace
the curve in the neighborhood of a point x = a by its rectilinear tangent

224 Analysis: Concerning Infinitesimal Calculus Proper.

just as, in physics and in other applications, we often discard the higher
powers of the independent variable in a series development (see Fig. 101).
In a similar manner we can obtain better approximations by making
use of parabolas of second, third, . . . order

y = A + Bx + Cx*, y = A+Bx + Cx* + Dx*> . . .

or, in analytic terms, by using polynomials of higher degree. Polynomials

are especially suitable because they are
so easy to calculate. We shall give all
these curves a special position, so that
at the point x = a they lie as close as
possible to the curve, i.e., so that they
shall be parabolas of osculation. Thus
the quadratic parabola will coincide with
y = f (x) not only in its ordinate but also
in its first and second derivatives (i.e.,

it will "osculate"). A simple calculation shows that the analytic ex-
pression for the parabola having osculation of order n will be

y = /M + ^(*-.)+^

(n = l,2,3, .-.)
and these are precisely the first n + 1 terms of Taylor's series.

The investigation as to whether and how far these polynomials
represent usable curves of approximation will be started by a some-
what experimental method, such as we used in the case (p. 194) of the
trigonometric series. I shall show you a few drawings of the first
osculating parabolas of simple curves, which were made 1 by Schimmack.
The first are the four following functions, all having a singularity at
x = \ , drawn with their parabolas of osculation at x = (see Figs. 102,
103, 104, 105).

1. lOg(1+*) W X- ~+ y- + .-. ,

X v2 v3

2. (1+*)* *M+ 8 -+ J6- + -,

3. (1+tf)- 1 ^!- x+ op y* H

4. (1 + #)- 2 ^l 2x + 3# 2 4* 3 H

In the interval ( 1 , +1) the parabolas approach the original curve
more and more as the order increases ; but to the right of x = +1 they
deviate from it increasingly, now above, now below, in a striking way.

1 Four of these drawings accompanied Schimmack's report on the Gottingen
Vacation Course, Easter, 1908: Uber die Gestaltung des mathematischen Unter-
richts im Sinne der neueren Reformideen, Zeitschrift fur den Mathematischen und
naturwissenschaftlichen Unterricht, vol. 39 (1908), p. 513; also separate reprints.
Leipzig, 1908.

Taylor's Theorem.

225

At the singular point x = \ , in Cases 1,3,4, where the original
function becomes infinite, the ordinates of the successive parabolas are
increasingly large. In Case 2, where the branch of the original curve
which appears, in the drawing, ends in # = 1 at a vertical tangent,

Fig. 102.

F"ig. 104.

all the parabolas extend beyond this point but approach the original
curve more and more at x = \ , by becoming ever steeper. At the
point x = +1 , symmetrical to % = 1 , the parabolas in the first two
cases approach the original curve more and more closely. In Case 3,
their ordinates are alternately equal to 1 and , while that of the original
curve has the value . In Case 4, finally, the ordinates increase in-
definitely with the order, and alternate in sign.

Klein, Elementary Mathematics. 1 5

226

Analysis: Concerning Infinitesimal Calculus Proper.

We shall examine, now, sketches of the osculating parabolas of

two integral transcendental functions (see Fig. 106, 107)

v 2 /8

s ,

6. si

3-1-^ --.-..

You notice that as their order increases, the parabolas give usable
aproximations to the original curve for a greater and greater interval.
It is especially striking in the case of sin x how the parabolas make
the effort to share more and more oscillations with the sine curve.
I call your attention to the fact that the drawing of such curves
in simple cases is perhaps suitable material even for the schools. After

we have thus assembled our experimental
I 1 material we must consider it mathematically.

Fig. 106.

Fig. 107-

The first question here is the extremely important one in practice as
to the closeness with which the w-th parabola of osculation represents
the original curve. This implies an estimate of the remainder for the
values of the ordinate, and is connected naturally with the passage
of n to infinity. Can the curve be represented exactly, at least for a part
of its course, by an infinite power series?

It will be sufficient to state the commonest of the theorems con-
cerning the remainder:

*/ -r j, / \i -r . T (n- 1)| ' v

The proof of the theorem is given in all the books and I shall revert
to it later, anyway, from a more general standpoint. The theorem is:
There is a value ( between a and x such that R n can be represented in the
form

*(*) =

' /<">(?),

(a <<*)..

Taylor's Theorem.

227

Fig. 108.

The question as to the justification of the transition to an infinite
series is now reduced to that as to whether this R n (x) has the limit
or not when n becomes infinite.

Returning to our examples, it appears, as you can verify by reading
anywhere, that in Cases 5 and 6 the infinite series converges for all
values of x. In Cases 1 to 4, it turns out that the series converges,
between \ and +1, to the original function, but that it diverges
outside this interval. For x = \ we have, in Case 2, convergence to
the function value; in Cases 1, 3, 4, the limiting value of the series as
well as that of the function is infinite, so that one could speak of con-
vergence here also, but it is not customary to use this word with a
series that has a definitely infinite limit. For y

x = +1, finally, we have convergence in the
first two examples, divergence in the last two.
All this is in fullest agreement with our graphs.

We may now raise the question, as we

did with the trigonometric series, as to the

limiting positions toward which the approxi-
mating parabolas converge, thought of as com-
plete curves. They cannot, of course, break
off suddenly at x = i 1 . For the case of
log (1 + x) I have sketched for you the limit
curve (Fig. 108). The even and odd parabolas
have different limiting positions, (indicated in the figure by dashes
and dots) which consist of the logarithm curve between 1 and +1
together with the lower and upper portions, respectively, of the
vertical line x = + 1 . The other three cases are similar.

The theoretical consideration of Taylor's series cannot be made com-
plete without going over to the complex variable. It is only then that
one can understand the sudden ceasing of the power series to converge
at places where the function is entirely regular. To be sure, one might
be satisfied, in the case of our examples, by saying that the series
cannot converge any farther to the right than to the left, and that the
convergence must cease at the left because of the singularity at x = \.
But such reasoning would not fit a case like the following. The Taylor's
series development for the branch of tan" 1 * which is regular for all
real x

tan- 1 *^* " + ~E h"-

converges only in the interval ( 1, + 1), and the parabolas of oscula-
tion converge alternately to two different limiting positions (see Fig. 109)^
The first consists, in the figure, of the long dotted parts of the vertical
lines x = +1, #= l together with the portion of the inverse tangent
curve lying between these verticals. The second limiting position is

15*

228

Analysis: Concerning Infinitesimal Calculus Proper.

Fig. 109-

obtained from the first by taking the short dotted parts of the vertical
lines instead of the long dotted parts. The convergence is toward the

first of these limit curves when we take
an odd number of terms in the series,
toward the second when we take an
even number. In the figure, the long dott-
ed curve represents y = x # 3 /3 + # 5 /5 ,
the short dotted curve is y = x # 3 /3 .
The sudden cessation of convergence at
~~jc the thoroughly regular points x = 1 is
incomprehensible if we limit ourselves
to real values of x and notice the be-
havior of the function. The explanation
is to be found in the important theorem
on the circle of convergence, the most
beautiful of Cauchy's function-theoretic
achievements, which can be stated as
follows. // one marks on the complex
x plane all the singular points of the analytic junction f (x) , when f (x) is
single-valued, and on the Riemann surface belonging to f (x) when f (x) is
many -valued, then the Taylor's series corresponding
to a regular point x = a converges inside the largest
circle about a which has no singular point in its
interior (i.e., so that at least one singular point
lies on its circumference). The series converges
for no point outside this circle (see Fig. 110).
Now our example tan" 1 A; has, as you know,
singularities at x = i, and the circle of con-
vergence of the development in powers of x is
consequently the unit circle about x = . The
convergence must cease therefore at x = 1 , since the real axis leaves
the circle of convergence at these points (see Fig. 111).

Finally, as to the convergence of the series on the
unit circle itself, I shall give you the reference which
came up when we were talking about the connection
between power series and trigonometric series. The
'* 7 convergence depends upon whether or not the real and
the imaginary part of the function, in view of the
singularities that must exist on the circle of conver-
gence, can be developed there into a convergent tri-
gonometric series.

I should like now to enliven the discussion of Taylor's theorem by
showing its relations to the problems of interpolation and of finite
differences. There, also, we are concerned with the approximation to

Fig. 110.

Fig. 111.

Taylor's Theorem.

229

Fig. 112.

a given curve by means of a parabola. But instead of trying to make
the parabola fit as closely as possible at one point, we require it to cut
the given curve in a number of preassigned points; and the question
is, again, as to how far this interpolation parabola gives a tolerable
approximation. In the simplest case, this amounts to replacing the curve
by a secant instead of the tangent (see Fig. 112). Similarly one passes
a quadratic parabola through three points
of the given curve, then a cubic parabola
through four points, and so on.

This is a natural way of approaching
interpolation, one that is very often em-
ployed, e.g., in the use of logarithmic
tables There we assume that the logarithmic curve runs rectilinearly
between two given tabular values and we interpolate "linearly" in the
well known way, which is facilitated by the difference tables. If this
approximation is not close enough, we apply quadratic interpolation.

From this broad statement of the general problem, we get a deter-
mination of the osculating parabolas in Taylor's theorem as a special
case, that is, when we simply allow the intersections with the inter-
polation parabolas to coincide. To be sure, the replacing of the curve
by these osculating parabolas is not properly expressed by the word
"interpolation" , except that one includes "extrapolation" in the problem
of interpolation. For example, the curve is compared not only with
the part of the secant lying between its points of intersection, but also
with the part beyond. For the entire pro-
cess the comprehensive word approximation
seems more suitable.

I shall now give the most important
formulas of interpolation. Let us first de-
termine the parabolas of order n \ which

cut the given function in the points x = a lt a 2 , . . ., a n , that is, whose
ordinates in these points are f(a^ } /(# 2 ), . . ., f(a n ) (see Fig. 113). This
problem, as you know, is solved by Lagrange's interpolation formula

(L f CU Z

Fig. 113.

y =

+
+

(x - a z ) (x - a 3 )

- ^2) K - 8 )

(x aj) (x - a 3 )

- a x ) (a 2 - a 8

The

It contains n terms with the factors / (a^ , / (0 2 ) ,,
numerators lack in succession the factors (x a x ) , (x a 2 ) , . . . ,
(x a n ) . It is easy to verify the correctness of the formula. For,
each summand of y, and hence y itself, is a polynomial in x of degree
n \ . If we put x = #! all the fractions vanish except the first, which

230 Analysis; Concerning Infinitesimal Calculus Proper.

reduces to 1 , so that we get y = / (a^ , Similarly we get y = f (a 2 ) for
% = 2 , etc.

From this formula it is easy to derive, by specialization, one that is
often called Newton's formula. This has to do with the case where
the abscissas a lt . . ., a n are equidistant (see Fig. 114). As the notation
* of the calculus of finite differences is advan-
/ tageous here we shall first introduce it.

Let Ax be any increment of x and let Af(x)
be the corresponding increment of f(x) so that

Now Af(x) is also a function of x which, if

we change x by Ax, will have a definite difference called the second
difference, A*f (x) t so that

In the same way we have

A*f(x + Ax) = A*f(x) + A*f(x) , etc.

This notation is precisely analogous to that of differential calculus,
except that one is concerned here with finite quantities and there is
no passing to the limit.

From the above definitions of differences there follows at once for
the values of / at the successive equidistant places

f(x+ Ax) =f(x) + Af(x),

f(x + 2Ax) = f(x + Ax) + Af(x + Ax)

(2)

f(x + 4Ax) = f(x) + 4Af(x) + 6A*f(x) + 4A*f(x) -

This table could be continued, the values at equidistant points being
expressed by means of successive differences taken at the initial point
and involving the binomial coefficients as factors,

Newton's formula for the interpolation parabola of order (n 1)
belonging to the n equidistant points of the x axis,

that is, which has at these points the same ordinates as / (x) , will be

f/ a \ I v* i "i v") i \ **/ \" * ""i tLJ-LL I
J / W i " -i | Ay. " o! (A*\* '

(3)

(x a) (x a A x) (x a (n

(-!)!

This is, in fact, a polynomial in * of order n 1 . For x = a it reduces

Taylor's Theorem. 231

to / (a); for x = a + Ax all the terms, except the first two, become
zero and there remains y == / (a) + Af (a), which by (2) is equal to
/ (a + Ax); and so on. Thus the table (2) yields a polynomial which
assumes the correct values at all the n places.

If we wish to use this interpolation formula to real advantage,
however, we must know something as to the correctness with which it
represents /(#), that is, we must be able to estimate the remainder.
Cauchy gave 1 the formula for this in 1840, and I should like to derive it.
I shall start from the more general Lagrange formula. Let x be any
value between the values a lf a 2 , . . ., a n , or beyond them (interpolation
or extrapolation). We denote by P (x) the ordinate of the interpolation
parabola given by the formula and by R (x) the remainder

(4) /(*) = P (*) + *(*).

According to the definition of P (x) the remainder R vanishes for

x = a lf a 2 , . . . , a n and we therefore set

R(x} = (jLz*)fr----(-*)y(a).

It is convenient to take out the factor n \ Then it turns out, in complete
analogy with the remainder term of Taylor's series, that \p (x) is equal
to the n-th derivative of f (x) taken for a value x = lying between the
n 1 points a lf a 2 , . . ., a n , x. This assertion that the deviation of / (x)
from the polynomial of order n 1 depends upon the entire course
of the function /< n ) (x) seems entirely plausible, if we reflect that / (x)
is equal to that polynomial when f^ (x) vanishes.

As to the proof of the remainder formula, we derive it by the following
device. Let us set up, as a function of a new variable z t the expression

where x remains as a parameter in v ; (x) . Now F (a^ = F (a 2 )
= F (a n ) = 0, since P (aj = / (aJ.P (a 2 ] = f (a,), . . ., P (a n ) = f (a n )
by definition. Furthermore F (x) = because the last summand goes
over into R (x), for z = x, so that the right side vanishes by (4). We
know, therefore, n + 1 zeros z = a lt a 2 , . . ., a n , x, of F(z). Now
apply ttie extended meap-value theorem, which one gets by repeated
application of the ordinary theorem (p. 213), namely: // a continuous
function, together with its first n derivatives, vanishes at n + 1 points,
then the n-th derivative vanishes at one point, at least, which lies in the
interval containing all the zeros. Hence if / (z) , and therefore also F (z) ,
has n continuous derivatives, there must be a value f between the
extremes of the values a l , a 2 , . . ; , , x for which

1 Comptes Rendus, vol. 11, pp. 775 789- CEuvres, 1st series, vol. 5, pp, 409
to 424, Paris, 1885-

232 Analysis: Concerning Infinitesimal Calculus Proper.

But we have

FM(z)=fW(z)-v(x),

since the polynomial P (z) of degree n 1 has for its n-ih derivative
and since only z n if (x)/n\ f the highest term of the last summand, has
an n-th derivative which does not vanish. Therefore we have, finally

vw = > r v.w :

which we wished to prove.

I shall write down Newton's interpolation formula with its remainder

term

(x a) (x a Ax)

(5)

_
-jT- -37- 2!

(AT- a) \x-a- (n-2)Ax]
+ " (n- 1)!

(*-fl) [*-0- (n-

where f is a mean value in the interval containing the n 1 points a .
a + Ax, a + 2 <4tf , ..., + ( !) ^#, #. The formula (5) is, in fact,
indispensable in the applications. I have already alluded to linear inter-
polation when logarithmic tables are used. If / (x) = log x and n = 2,
we find, from (5)

, , . x a Aloga (x a) (x a Ax) M

log* = loga + _ -- - [ - -- ^ ---- i - ? .

Since d* log x/dx^ = Mix 2 where M is the modulus of the logarithmic
system. Hence we have an expression for the error which we commit
when we interpolate linearly between the tabular logarithms for a and
a + A x . This error has different signs according as x lies between a
and a + Ax or outside this interval. Every one who has to do with
logarithmic tables should really know this formula.

I shall not devote any more attention to applications, but shall now
draw your attention to the marked analogy between the interpolation
formula of Newton and the formula of Taylor. There is a substantial
reason for this analogy. It is easy to give an exact deduction of Taylor's
theorem from the Newtonian formula, corresponding to the passage to
the limit from interpolation parabolas to osculating parabolas. Thus,
if we keep x, a, and n fixed and let A% converge to zero, then, since
/ (x) has n derivatives, the n 1 difference quotients in (5) go over
into the derivatives

Af(a) ,,, x ,. A 2 f(a) //// \

~ = - =

In the last term of (5), the value of f can change with decreasing Ax.
Since all the other terms on the right have definite limits, however,
and the left side has the fixed value / (x) during the entire limit process,
it follows that the values of / (n ^(l) must converge to a definite value

Taylor's Theorem. 233

and that this value, furthermore, must, because of the continuity of / (n) ,
be a value of this function for some place between a and x . If we denote
this again by I we have

/ W = /(*) + i=f (a) + - - - + ? /*-() + =

Thus we have obtained a complete proof of Taylor's theorem with the
remainder term and at the same time have given it an ordered place
in the theory of interpolation.

It seems to me that this proof of Taylor's theorem, which brings
it into wider relation with very simple questions and which provides
such a smooth passage to the limit, is the very best possible one. But
all the mathematicians to whom these things are familiar (it is remark-
able that they are unknown to many, including perhaps even some
writers of textbooks) do not think so. They are accustomed to confront
a passage to a limit with a very grave face and would therefore prefer
a direct proof of Taylor's theorem to one linking it with the calculus
of finite differences.

I must emphasize however that, as a matter of history, the source
of Taylor's theorem is actually the calculus of finite differences.
I have already mentioned that Brook Taylor first published it in his
Methodus incrementorum 1 . He first deduces Newton's formula, without
the remainder, of course, and then puts in it Ax = and n = oo. He
thus gets correctly from first terms of Newton's formula the first terms
of his new series:

The continuation of this series, according to the same law, seems to him
self evident, and he gives no thought either to a remainder term or to
convergence. We have here, in fact, a passage to the limit of unexampled
audacity. The earlier terms, in which x a Ax, x a 2Ax, ...
appear, offer no difficulty, because these finite multiples of A x approach
zero with A x ; but with increasing n there appear terms in ever increasing
number, ^presenting factors x a kAx with larger and larger k, and
.one is not justified in treating these forthwith in the same way and in
assuming that they go over into a convergent series.

Taylor really operates here with infinitely small quantities (differen-
tials) in the same unquestioning way as the Leibnizians. It is interesting
to reflect that although, as a young man of twenty-nine, he was under
the eye of Newton, he departed from the latter's method of limits.

You will find an excellent critical presentation of the entire develop-
ment of Taylor's theorem in Alfred Pringsheim's memoir: Zur Geschichte

1 Londini, 1715, p. 21-23-

234 Analysis: Concerning Infinitesimal Calculus Proper.

des Taylorschen Lehrsatzes 1 . I should like to speak here about the
customary distinction between Taylor's series and that of Maclaurin.
As is well known, many textbooks make a point of putting a = and
of calling the obvious special case of Taylor's series which thus arises:

/w = /(o) + ^f(o) + ~no)-+---

the series of Maclaurin ; and many persons may think that this distinction
is important. Anybody who understands the situation however sees
that it is comparitively unimportant mathematically. But it is not
so well known that, considered historically, it is pure nonsense. For
Taylor had undoubted priority with his general theorem, deduced in
the way indicated above. More than this, he emphasizes at a later
place in his book (p. 27) the special form of the series f or a and
remarks that it could be derived directly by the method which is called
today that of undetermined coefficients. Furthermore, Maclaurin took
over 2 this deduction in 1742 in his Treatise of Fluxions (which we
mentioned on p. 212) where he quoted Taylor expressly and made no
claim whatever of offering anything new. But the quotation seems to
have been disregarded and the author of the book seems to have been
looked upon as the discoverer of the theorem. Errors of this sort are
common. It was only later that people went back to Taylor and named
the general theorem, at least, after him. It is difficult, if not impossible,
to overcome such deeprooted absurdities. At best, one can only spread
the truth in the small circle of those who have historical interests.
I shall now supplement our discussion of infinitesimal calculus with
some remarks of a general nature.

3. Historical and Pedagogical Considerations

I should like to remind you, first of all, that the bond which Taylor
established between difference calculus and differential calculus held for
a long time. These two branches always went hand in hand, still in the
analytic developments of Euler, and the formulas of differential calculus
appeared as limiting cases of elementary relations that occur in the
difference calculus. This natural connection was first brokeh by the
oft mentioned formal definitions of Lagrange's derivative calculus'.
I should like to show you a compilation from the end of the eighteenth
century which, closely following Lagrange, brings together all the facts
then known about infinitesimal calculus, namely the Traite du Calcul
Differentiel et du Calcul Integral of Lacroix 3 . As a characteristic sample
from this work, consider the definition of the derivative (vol. I, p. 145)'-

1 Bibliotheca Mathematica, 3rd series, voL I (1900), p. 433 479-

2 Edinburgh, 1742, vol.11, p. 610.

3 Three volumes, Paris, 17971800, with many later editions.

Historical and Pedagogical Considerations. 2)5

A function / (x) is defined by means of a power series. By using the
binomial theorem (and rearranging the terms) one has

Lacroix now denotes the term of this series which is linear in h by df (x) ,
and, writing dx for h itself, he has for the derivative, which he calls diffe-
rential coefficient

Thus this formula is deduced in a manner thoroughly superficial even
if unassailable. Within the range of these thoughts, Lacroix could not,
of course, use the calculus of differences as a starting point. However,
since this branch seemed to him too important in practice to be omitted,
he adopted the expedient of developing it independently, which he did
very thoroughly in a third volume, but without any connecting bridge
between it* and differential calculus.

This "large Lacroix" is historically significant as the proper source
of the many textbooks of infinitesimal calculus which appeared in the
nineteenth century. In the first rank of these I should mention his
own textbook, the "small Lacroix" 1 .

Since the twenties of the last century the textbooks have been
strongly influenced also by the method of limits which Cauchy raised
to such an honorable place. Here we should first think of the many
French textbooks, most of which, as Cours d' Analyse de VEcole Poly-
technique, were prepared expressly for university instruction. Directly
or indirectly, German textbooks also have depended on them, with the
single exception, perhaps, of the one by Schlomilch. From the long
list of books, I shall single out only Serret's Cours deCalcul Difftrentiel
et Integral, which appeared first in 1869 in Paris. It was translated into
German in 1884 by Axel Harnack and has been since then one of our
most widely used textbooks. It suffered as to symmetry at the hands
of a long series of revisers. The editions 2 which have appeared since
1906, however, have been subjected to a thoroughgoing revision by
G. Scheffers of Charlottenburg, the result being a homogeneous work.
I am glad to mention also an entirely new French book, the Cours
d' Analyse Mathematique by Goursat 3 in three volumes, which is fuller
in many ways than Serret and contains, in particular, a long series of
entirely modern developments. Furthermore it is a very readable book.

1 TraM Ettmentaire du Calcul Difffrentiel et Integral. Two volumes, Paris, 1797.

2 Since 1906: Serret, J, A., u. G. Scheffers, Lehrbuch der Differential- und
Integralrechnung, vol. I, sixth edition. Leipzig 1915; vol. II, 67 edition; vol. Ill,
fifth edition, 1914.

8 Paris 19021907, vol. I, third edition. 1917; vol. II, third edition. 1918;
vol. Ill, second edition. 1915- (Translated into English: vol. I by E. R. Hedrick,
1904, Ginn and Co.; vol. II by E. R. Hedrick and O. Dunkel, 1916, Ginn and Co.)

Analysis: Concerning Infinitesimal Calculus Proper.

In all these recent books, the derivative and the integral are based
entirely upon the concept of limit. There is never any question as to
difference calculus or interpolation. One sees the thing in a clearer
light, perhaps, in this way, but, on the other hand, the field of view is
considerably narrowed, as it is when we use a microscope. Difference
calculus is now left entirely to the practical calculators, who are obliged
to use it, especially the astronomers; and the mathematician hears
nothing of it. We may hope that the future will bring a change 1 here.

As a conclusion of my discussion of infinitesimal calculus I should
like to bring up again for emphasis four points, in which my exposition
differs especially from the customary presentation in the textbooks:

1. Illustration of abstract considerations by means of figures (curves
of approximation, in the case of Fourier's and Taylor's series).

2. Emphasis upon its relation to neighboring fields, such as calculus
of differences and of interpolation, and finally to philosophical investiga-
tions.

3. Emphasis upon historical growth.

4. Exhibition of samples of popular literature to mark the difference
between the notions of the public, as influenced by this literature and those
of the trained mathematician.

It seems to me extremely important that precisely the prospective
teacher should take account of all of these. As soon as you begin teaching
you will be confronted with the popular views. If you lack orientation,
if you are not well informed concerning the intuitive elements of mathe-
matics as well as the vital relations with neighboring fields, if, above
all, you do not know the historical development, your footing will be
very insecure. You will then either withdraw to the ground of the
most modern pure mathematics, and fail to be understood in the school,
or you will succumb to the assault, give up what you learned at the
university, and even in your teaching allow yourself to be buried in
the traditional routine. The discontinuity between school and uni-
versity, of which I have often spoken, is greatest precisely in the field
of infinitesimal calculus. I hope that my words may contribute to its
removal and that they may provide you with useful armor in your
teaching.

This brings me to the end of the conventional analysis. By way of
supplement I shall discuss a few theories of modern mathematics to
which I have referred occasionally and with which I think the teacher
should have some acquaintance.

1 In order to make a beginning here, I induced Friesendorff and Prumm to
translate Markoffs Differenzenrechnung into German (Leipzig, 1896). There is
a series of articles in the Enzyklopadie. A work on Differenzenrechnung by E. Nor-
lund has just appeared (Berlin, Julius Springer, 1924) which exhibits the subject
in new light.

Supplement

I. Transcendence of the Numbers e and a

The first topic which I shall discuss will be the numbers e and n.
In particular, I wish to prove that they are transcendental numbers.

Interest in the number n, in geometric form, dates from ancient
times. Even then it was usual to distinguish between the problem of
its approximate calculation and that of its exact theoretical construction;
and one had certain fundamentals for the solution of both problems.
Archimedes made an essential advance, in the first, with his process of
approximating to the circle by means of inscribed and circumscribed
polygons. The second problem soon centered in the question as to
whether or not it was possible to construct n with ruler and compasses.
This was attempted in all possible ways with never a suspicion that the
reason for continued failure was the impossibility of the construction.
An account of some of the early attempts has been published by Rudio 1 .
The quadrature of the circle still remains one of the most popular
problems, and many persons, as I have already remarked, seek salvation
in its solution, without knowing, or believing, that modern science has
long since settled the question.

In fact, these ancient problems are completely solved today. One
is sometimes inclined to doubt whether human knowledge really can
advance, and in some fields the doubt may be justified. In mathematics,
however, there are indeed advances of which we have here an example.

The foundations upon which the modern solution of these problems
rests date from the period between Newton and Euler. A valuable tool
for the approximate calculation of n was supplied by infinite series, a
tool whith made possible an accuracy adequate for all needs. The
most elaborate result obtained was that of the Englishman Shanks, who
calculated n to 707 places 2 . One can ascribe this feat to a sportsmanlike
interest in making a record, since no applications -could ever require
such accuracy.

On the theoretical side, we find the number e, the base of the system
of natural logarithms, coming into the investigations during the same

1 Der Bericht des Simplicius uber die Quadraturen des Antiphon und Hippokrates.
Leipzig, 1908.

2 See Weber- Wellstein, vol. l, p. 523-

2^8 Supplement: Transcendence of the Numbers e and n.

period. The remarkable relation e in = 1 was discovered and a means
was developed in the integral calculus which, as we shall see, was of
importance for the final solution of the question as to the quadrature
of the circle. The decisive step in the solution of the problem was taken
by Hermite 1 in 1873, when he proved the transcendence of e. He did
not succeed in proving the transcendence of n. That was done by
Lindemann 2 in 1882.

These results represent an essential generalization of the classical
problem. That was concerned only with the construction of n by means
of ruler and compasses, which amounts, analytically, as we saw (p. 51)
to representing n by a finite succession of square roots and rational
numbers. But the modern results prove not merely the impossibility
of this representation; they show far more, namely, that n (and like-
wise e) is transcendental, that is, that it satisfies no algebraic relation
whatever whose coefficients are integers. In other words, neither e
nor n can be the root of an algebraic equation with integral coefficients :

a Q + ax + a 2 x 2 + - - + a n x n =

no matter how large the integers a Q> . . ., a n or the degree n. It is
essential that the coefficients be integers. It would suffice however to
say fractions, since we could make them integral by multiplying through
by a common denominator.

I pass now to the proof of the transcendence of e, in which I shall
follow the simplified method given by Hilbert in Volume 43 of the
Mathematische Annalen (1893). We shall show that the assumption
of an equation

(1) a + a^e + a 2 e 2 + - - - + a n e n = 0, where a -+ 0,

in which , . . .,a n are integers, leads to a contradiction. This will
involve the use of only the simplest properties of whole numbers. We
shall need, namely, from the theory of numbers, only the most ele-
mentary theorems on divisibility, in particular, that an integer can be
separated into prime factors in only one way, and, second, that the
number of primes is infinite.

The plan of the proof is as follows. We shall set up a method which
enables one to approximate especially well to e and powers of e, by.
meajis of rational numbers, so that we have

M * _ M l+ g l ,2 _ M 2 + *2 _ W _ M n + *V

e - ' e - ' ' ' " e ~ ~~

where M , M lt M 2 , . . ., M n are integers, and t /M, e 2 /M, . . ., n /M are

1 Comptes Rendus, vol. 77 (1873), p. 18-24, 74-79, 226-233, 285-293;
== Werke III (1912), p. 150.

2 Sitzungsberichte der Berliner Akademie, 1882, p. 679. and Mathematische
Annalen, vol.20 (1882), p. 213-

Transcendence of e.

239

very small positive fractions. Then the assumed equation (1), after
multiplication by M, takes the form

(3) [a Q M+a l M l + a^M 2 -\ ----- h *MJ + \a^+ a^-\ ----- Va n e n } =0.

The first parenthesis is an integer, and we shall prove that it is not
zero. As for the second parenthesis, we shall show that e lt . . ., e n can
be made so small that it will be a positive proper fraction. Then we
shall have the obvious contradiction that an integer a Q M + a l M l +
+ a n M n which is not zero, increased by a proper fraction a^ +
+ a n e n is zero. This will show the impossibility of (1).

In the course of the discussion which I have just outlined we shall
make use of the theorem that if an integer is not divisible by a definite
number, the integer cannot be zero (for zero is divisible by every number).
We shall show, namely, that M x , . . . , M n are divisible by a certain
prime number p, but that a Q M does not contain p, and that, therefore,
a Q M + a l M l + + a n M n is not divisible by p, and hence is different
from zero.

The principal aid in carrying out the indicated proof comes from
a certain definite integral which was devised by Hermite for this pur-
pose and which we shall call Hermite' s integral. The key to this proof
lies in its structure. This integral, whose value, as we shall see, is a
positive whole number and which we shall use to define M , is

~ -.

(P l) !

where n is the degree of the assumed equation (1), and p is an odd
prime which we shall determine later. From this integral we shall get
the desired approximation (2) to the powers e v (v = 1 , 2 , . . . , n) by
breaking the interval of integration of the integral M e v at the point v
and setting

Uti M - * r* P ~ l[( *~ 1) "- ('-*)]'*"' dz

(4a) Mv ~ e } v (f=W '

/,u\ v - ... - .

(4b) , e v = e' ______ dz .

Let us now take up the details of the proof.
1. We start with the well known formula from the beginnings of
the theory of the gamma function:

f
Jo

We shall need this formula only for integral values of Q , in which case
= (Q 1)!, and I shall deduce it under this restriction. If we

240

Supplement: Transcendence of the Numbers e and a.

integrate by parts we have, for Q > 1 :

["
Jo

= (Q i)fze- 2 e- z dz.
Jo

The integral on the right is of the same form as the one on the left,
except that the exponent of z is reduced. If we apply this process
repeatedly we must eventually come to 2, since Q is an integer; and

/oo

since / e~ z dz = 1 , we obtain finally

(5) r#-*

= ( e - i) ( e - 2) . . . 3 2 i = (e - 1) i

Thus for integral Q the integral is a whole number which increases
very rapidly when Q increases.

To make this result clear geometrically, let us draw the curve
y = z Q ~ l e~ z for different values of Q. The value of the integral will
then be represented by the area under the curve extending to infinity

(see Fig. 115). The larger

* Q is the more closely the

curve hugs the z axis at
the origin, but the more
rapidly it rises beyond
z = 1 . The curve has a
maximum at z = Q 1 ,
for all values of g; in
other words the maximum
occurs farther and farther

to the right as Q increases ; and its value also increases with Q . To the
right of the maximum, the factor e~ z prevails so that the curve falls,
approaching the z axis asymptotically. It is thus comprehensible that
the area (our' integral) always remains finite but increases rapidly with Q.
2. With this formula we can now easily evalute our Hermite integral.
Developing the integrand by the polynomial theorem

1 z 3 14 5

Fig. 115.

where only the terms involving the highest and the lowest powers of z
have been written down, the integral becomes

np+p

Transcendence of e. 241

The C Q are integral constants, by the polynomial theorem. Now we
can apply formula (5) to each of these integrals and obtain

np+p

The summation index Q is always larger than p and consequently
(Q !)!/( 1)! is an integer and one which contains p as a factor,
so that we can take p as a factor out of the entire sum:

Now, so far as divisibility by p is concerned, M must behave like
the first summand ( l) n (n!) p . And since p is a prime number it will
not be a divisor of this summand if it is not a divisor of any of its factors
1 , 2 , . . . , n, which will certainly be the case if p > n . But this condition
can be satisfied in an unlimited number of ways, since the number of
primes is infinite. Consequently we can bring it about that ( \) n (n\) p ,
and hence M, is not divisible by p.

Since furthermore a Q =)= 0, we can see to it, at the same time, that
fl is not divisible by p by selecting p larger also than |a |, which is,
of course, possible, by what was said above. But then the product
M will not be divisible by p , and that is what we wished to show.

3. Now we must examine the numbers M v (v = \ , 2, . . . , n) , defined
in (4 a) (p. 239). Putting the factor e v under the sign of integration and
introducing the new variable of integration f = z v , which varies
from to oo when z runs from v to oo, we have

_ r
"Jo

This expression has a form entirely analogous to the one considered
before for M and we can treat it in the same way. If we multiply out
the factors of the integrand there will result an aggregate of powers
with integral coefficients of which the lowest will be f. The integral
of the numerator will thus be a combination of the integrals

fFc-ed, rV^-^f, ..., r
Jo Jo Jo

and since these are, by (5), equal to p\,(p + 1)!, ... the numerator
will be p I multiplied by a whole number A , so that we have

M * = - =#-4,, (" = 1, 2, . . . , n) .

In other words, every M v is a whole number which is divisible by p.
This, combined with the result of No. 2, proves the statement made
on p. 239 that a M + a l M l + + a n M n is not divisible by p and
is therefore different from zero.

242 Supplement: Transcendence of the Numbers e and n.

4. The second part of the proof has to do with the sum a^ s l +
+ a n e n , where, by (4b),

" 1 [(^- l)(*-2) ...(*-

v ~jo

We must show that these e v can be made arbitrarily small by an appro-
priate choice of p . To this end we use the fact that we can make p as
large as we chose; for the only conditions thus far imposed upon p are
that it should be a prime number larger than n and also larger than
|0 |, and these can be satisfied by arbitrarily large prime numbers.
Let us examine the graph of the integrand. At z = it will be
tangent to the z axis, but at z = 1 , 2, . . . , n (in Fig. 116, n = 3) it

will be tangent to the
z axis and also cut it,
since p is odd. As we
shall see soon, the
presence in the deno-
minator of (p 1)!
brings it about that
for large p the curve
y ' * * * departs but little

Fig. 116. , from the z axis in the

interval (0, n), so

that it seems plausible that the integrals s v should be very small.
For z > n the curve rises and runs asymptotically like the former
curve z e ~ l e~ z [iQi Q = (n + \)p] and finally approaches the z axis. It
was for this reason that the value M of the integral (when the interval
of integration was from to oo) increased so rapidly with p.

In actually estimating the integrals we can be satisfied with a rough
approximation. Let G and g v be the maxima of the absolute values
of the functions z (z 1) ... (z n) and (z 1) (z 2) ... (z n) e~ z+v
respectively in the interval (Q,ri):

Since the integral of a function is never larger than the integral of its
absolute value, we have, for each E V

Now G, g v , and v are fixed numbers independent of p, but the number
(/> !)! in the denominator increases ultimately more rapidly than
the power G p ~\ or, more exactly, the fraction G p ~ l /(p 1)! becomes,
for sufficiently large p, smaller than any preassigned number, however
small. Thus, because of (6), we can actually make each of the n numbers s v
arbitrarily small by choosing p sufficiently large.

Transcendence of a. f ' 243

It follows immediately from this that we can also make the sum of n
terms a e l + + a n e n arbitrarily small. We have, in fact

and by (6)

/i i j i i i /* i .it \ G p-1

^(lir i'fo + li|-2gi+ + W-w-gn) -77 n

Since the parenthesis has a value which is independent of p, we can,
by virtue of the factor G p ~ l /(p 1)!, make the entire right hand side,
and hence also a e + a 2 e 2 + + #n n, as small as we choose, and,
in particular, smaller than unity.

With this we have shown, as we agreed to do (p. 239), that the.
assumption of the equation (3)

leads to a contradiction, namely that a non vanishing integer increased
by a proper fraction gives zero. And since this equation cannot exist
the transcendence of e is proved.

Proof of the Transcendence of n

We turn now to the proof of the transcendence of the number n.
This proof is somewhat more difficult than the foregoing, but it is still
fairly easy. It is only necessary to begin at the right end, which is
indeed the art of all mathematical discovery. '

The problem, as Lindemann considered it, was the following: It has

been shown thus far that an equation ^?a v e v = cannot exist if the

coefficients a v and the exponents v of e are ordinary whole numbers.
Would it not be possible to prove a similar thing where a v and v are
arbitrary algebraic numbers? He succeeded in doing this; in fact, his most
general theorem concerning the exponential function is as follows: An

equation^a v e b v = cannot exist if the a v , b v are algebraic numbers, whereby

the a v are arbitrary, the b v different from one another. The transcendence
of n is then a corollary to this theorem. For, as is well known, 1 + & in ;
and if n were an algebraic number, i n would be also, and the existence
of this equation would contradict the above theorem of Lindemann.
I shall now prove in detail only a certain special case of Linde-
mann's theorem, one which carries with it, however, the transcendence
of n. I shall follow again, in the main, Hilbert's proof in Volume 43 of
the Mathematische Annalen, which is essentially simpler than Linde-
mann' s, and which is an exact generalization of the discussion which
we have given for e.

244 Supplement: Transcendence of the Numbers e and a.

The starting point is the relation
(1) 1 + e in = 0.

If, now, n satisfies any algebraic equation with integral coefficients
then in also satisfies such an equation. Let a l9 <x 2 , . . ., <x n be all the
roots, including i n itself, of this last equation. Then we must also
have, because of (1):

(1 + e**)(\ +^ a ) (1 + *") = ().
Multiplying out we obtain

e 01 ^"* -{- - + e**~~ l+ **)

(2)

1 ' * -j- (!+*+ +**) = 0.

Now some of the exponents which appear here might, by chance, be
zero. Everytime that this occurs the left hand sum has a positive
summand 1 , and we combine these, together with the 1 that appears
formally, into a positive integer , which is certainly different from
zero. The remaining exponents, all different from zero, we denote by
Pi> @2> > PN and we write, accordingly, instead of (2) ,

(3) a v + ^ + ^ + - + eP N = 0, where > .

Now Pi, . . ., (} N are the roots of an algebraic equation with integral
coefficients. For, from the equation whose roots are a x , . . ., oc n we
can construct one of the same character whose roots are the two term
sums x + 2 a i + a s t* 1611 another for the three term sums

#1 + ^2 + a a> a i + ^2 + *4f an d so on; finally, 04 + <* 2 H 1- a n

is itself rational and satisfies therefore a linear integral equation. By
multiplying together all these equations, we obtain again an equation
with integral coefficients, which might have some zero roots, and whose
remaining roots are /8 1 , . . ., fi N . Omitting the power of the unknown
which corresponds to the zero roots there will remain for the N quanti-
ties /8 an algebraic equation of degree N with integral coefficients and
absolute term different from zero

(4) b + b^z + b z z* + + b N z N = 0, where 6 0f b N =j= 0.

We now have to prove the following special case of Liqdemann's
theorem. An equation of the form (3), with integral non-vanishing 0t ,
cannot exist if fi^, . . . , (} N are the roots of an algebraic equation of degree N,
with integral coefficients. This theorem includes the transcendence of n.

The proof involves the same steps as the one already given for the
transcendence of e. Just as we could there approximate closely to the
powers e 1 , e 2 , . . ., e n by means of rational numbers, so we shall be
concerned here with the best possible approximation to the powers
of e which appear in (3), and we shall write., in the old notation,

lt\ tfi M I + S I ^ _ M Z +
W * - M ' * ~ M

Transcendence of si. 245

where the denominator M is again an ordinary integer but M l , . . . , M N
are not integers as formerly, but are integral algebraic numbers, and
the /?!, . . ., fa, which in general can now be complex, are in absolute
value very small. It is here that the difficulty in this proof lies, as
compared to the earlier one. The sum of all the Af lf . . . , M N will again,
however, be ah integer, and we shall be able to arrange it so that the
first summand in the equation:

(6) [a M + M l + M z + + My] + [fi x + f. + + *N\ = 0,

[into which (3) goes over when we multiply by M and use (5)] will be
a non-vanishing integer, while the second summand will be, in absolute
value, smaller than unity. Essentially, this will be the same contradiction
which we used before. It will show the impossibility of (6) and (3)
and so complete our proof. As to detail, we shall again show that M l
+ M 2 + + M N is divisible by a certain prime number p , but that
a Q M is not, which will show that the first summand in (6) cannot
vanish; then we shall choose p so large that the second summand will
be arbitrarily small.

1. Our first concern is to define M by a suitable generalization of
Hermite's integral. A hint here lies in the fact that the zeros of the
factor (z 1) (z 2) ... (z n) in Hermite's integral were the ex-
ponents of e in the hypothetical algebraic equation. Hence we now
replace that factor by the product made by using the exponents in (3),
i.e., the solutions in (4):

(7) (z - A)(* - A) ...(*-&)= [6 + M +

It turns out to be essential here to put in a suitable power of b N as
factor, which was unnecessary before because (z 1) ... (z n) was
integral. We set then finally

(8) M

2. Just as before, we now develop the integrand of M according to
powers of z. The lowest power, that belonging to z p ~ l , gives then:

where the integral has been evaluated by means of the gamma formula
(p. 239). The remaining summands in the integrand contain either z p
or still higher powers, so that the integrals contain the factor p\/p 1) ! ,
multiplied by integers, and are thus all divisible by p . Consequently M
is an integer which is certainly not divisible by p, i.e., provided the
prime number p is not a divisor of either 6 or b$ . But since these two
numbers are both different from zero, we can bring this about by
choosing p so that p > |6 | and also p > \b N \.

246 Supplement: Transcendence of the Numbers e and n.

Since a > it follows that a Q M is not divisible by p if we impose
the additional condition p > . Inasmuch as the number of primes
is infinite we can satisfy all these conditions in an unlimited number
of ways.

3. We must now set up M v and e v . Here we must modify our
earlier plan because the p v , which now take the place of the old v , can
be complex; in fact one them is in. If we are to split the integral M
as we did before we must first determine the path of integration in the
complex plane. Fortunately the integrand of our integral is a finite
single-valued function of the variable of integration z, regular every-
where except at z = oo, where it has an essential singularity. Instead
of integrating from to oo along the real axis we can choose any other
path from tooo, provided it ultimately runs asymptotically parallel
/\ to the positive real half axis. This is

2-Plan6 /' \ necessary if the integral is to have a

meaning at all, in view of the behavior
of e~ z in the complex plane.

Let us now mark the N points ft l9
Pz, > , PN in the plane and recall
that we shall obtain the same value

2 for M if we first integrate rectilinearly

from to one of the points fa and
Fi then to oo along a parallel to the real

axis (see Fig. 117). Along this path
we can separate M into the two characteristic parts: The rectilinear
path from to fa supplies the e v which will become arbitrarily small
with increasing p; the parallel from fa to oo will supply the integral
algebraic number My:

(8a) , = > -~ [6. + b,z +

(v = 1, 2, . . . , N) ,

//oo z p-l J
" ' - Jr. V-i)V [6 + * * + + b zN} * b ~ 1]p ~ l

These assumptions satisfy (5). Our choice of a rectilinear path of
integration was made solely for convenience; a curvilinear path from
to ft v would, of course yield the same value for e v , but it is easier to
estimate the integral when the path is straight. Similarly, we might
choose, instead of the horizontal path from fi v to oo, an arbitrary curve
provided only that it approached the horizontal asymptotically; but
that would be unnecessarily inconvenient.

4. I will discuss first the estimation of the e v , because this involves
nothing new if we only recall that the absolute value of a complex
integral cannot be larger than the maximum of the absolute value of
the integrand, multiplied by the length of the path of integration,

Transcendence of n. 247

.which, in our case, is | f{ v \ . The upper limit for e v would be, then, the
product of G p ~ li l(p 1) ! by factors which are independent of p t where G
denotes the maximum of \z(b$ + bz + + b N z^) b%~ l \ in a region
which contains all the segments joining with the ft v . From this one
may infer, as we did before, (p. 243), that, by sufficiently increasing p,
the value of each e v and, therefore, the value of e l + + S N can be
made as small as we please and, in particular, smaller than unity.

5. It is only in the discussion of the M v that essentially new con-
siderations enter, and these are, to be sure, only generalizations of our
former reasoning, due to the fact that integral algebraic numbers take
the place now of what were then integral rational numbers. We shall
consider, as a whole, the sum:

If we make use of (7) (p. 245) and replace, in each summand of the
above summation, the polynomial in z by the product of the factors
( z ~ Pi) '" ( z &0 an d introduce the new variable of integration
C = z /?, which will run through real values from to oo, we obtain

N N

(9) 1-1

/*00 g_^

which may be written = / ,. __ >, p $ (t) ,

where we use the abbreviation

This sum ^P(f), like each of its N summands, is a polynomial in f .
In each of the summands, one of the N quantities jS lt . . ., f} N plays a
marked role; but if we consider the polynomial in f obtained by multi-
plying out in $(?), we see that these N quantities appear, without
preference, in the coefficients of the different powers of . In other
words, each of these coefficients is a symmetric function of ^ , . . . , fa .
'The multiplying out of the individual factors by the multinomial theorem
permits the further inference that these functions /? x , . . . , fa are rational
integral functions with rational integral coefficients. But according to
a well known theorem in algebra, any rational symmetric function, with
rational coefficients, of all the roots of a rational equation is itself
always a rational number; and since the &, . . . 9 fa are all the roots
of equation (4), the coefficients of #() are actually rational numbers.

But, ipore than this, we need rational integral numbers. These are
supplied by the power of by which occurs as a factor of (f). We can,

248 Supplement: Transcendence of the Numbers e and a.

in fact, distribute this power among all the linear factors which occur
there and write

(9")

In analogy with what we had earlier, the coefficients of f , when this
polynomial is calculated, are rational integral symmetric functions of
the products by Pi, b N j3 2 , . . ., b N fi N , with rational integral coefficients.
But these N products are roots of the equation into which (4) goes if
we replace z by z/by:

If we multiply through by b% 1 this equation goes over into:

(10) W'+ &i &$-** + + by-tbyZ*'* + by.iZ*-* + Z N = 0,

that is, an equation with integral coefficients when the coefficient of
the highest power is unity. Numbers which satisfy such an equation
are called integral algebraic numbers, and we have the following refinement
of the theorem mentioned above : Rational integral symmetric functions,
with rational integral coefficients, of all the roots of an integral equation
whose highest coefficient is unity (i.e., of integral algebraic numbers) are
themselves rational integral numbers. You will find this theorem in text-
books on algebra; and if it is not always enunciated in this precise
form you can, nevertheless, by following the proof, convince yourselves
of its correctness.

Now the coefficients of the polynomial <&() actually satisfied the
assumptions of this theorem so that they are rational integral numbers
which we shall denote by A Q , A lt . . ., Ay p -i. We have, then,

With this we have essentially reached our goal. For, if we carry
out the integrations in the numerator, using our gamma formula (p. 239) >
we obtain factors p\, (p + 1) I, (p + 2)1 . . ., since each term contains
as factor a power of pot degree p or higher; and after division by (p 1) !
there remains everywhere as factor a multiple of p, while the other

factors are rational integral numbers (the A Q , AI, A 2 , . . .). Thus ^M v
is certainly a rational integral number divisible by p. re=1

We saw (p. 246) that a M was not divisible by p , so that

v=l

Transcendence of n. 249

is necessarily a rational integral number which is not divisible by p and
hence, in particular, different from zero. Therefore the equation (6) :

cannot exist, for a non vanishing integer added to ^?e v , which was

shown in No. 4 (p. 247) to be smaller than unity in absolute value,
cannot yield zero. But this proves the special case of Lindemann's
theorem which we enunciated above (p. 244) and which carries with it
the transcendence of n.

I should like to mention here another interesting special case of
the general Lindemann theorem, namely, that in the equation 0^ = b
the numbers 6, /? cannot both be algebraic, with the trivial exception
/? = 0, b = 1 . In other words, the exponential function of an algebraic
argument (i as well as the natural logarithm of an algebraic number b
is, with this one exception, transcendental. This statement includes the
transcendence of both e and n , the former for /? = 1 , the latter for
b = \ (because e i71 = 1) . The proof of this theorem can be effected
by an exact generalization of the last discussion. One would start
from b eP instead of from 1 + e" as before. It would be necessary
to take into account not only all the roots of the algebraic equation
for {{, but also all the roots of the equation for b, in order to arrive
at an equation analogous to (3), so that one would need more notation
and the proof would be apparently less perspicuous; but it would require
no essentially new ideas.

I shall not go farther into these proofs, but I should like to point
out graphically the significance of the last theorem concerning the ex-
ponential function. Let us think of all points with an algebraic abscissa

as marked off on the % axis ff. >JC . We

know that even the rational numbers, and hence, with greater reason,
the algebraic numbers lie everywhere dense on the x axis. One might
think at first that the algebraic numbers would exhaust the real numbers.
But our theorem declares that this is not the case; that between the
algebraic numbers there are infinitely many other numbers, viz. the
'transcendental numbers; and that we have examples of them in unlimited
quantity in algbr - no -, in log (algebr.no.), and in every algebraic function
of these transcendental numbers. It will be more obvious, perhaps, if
we write the equation in the form y = e x and draw the curve in an
x y plane (see Fig. 118). If we now mark all the algebraic numbers on
the x axis and on the y axis and consider all the points in the plane
that have both an algebraic x and an algebraic y , the x y plane will be
"densely" covered with these algebraic points. In spite of this dense
distribution, the exponential curve y = f does not contain a single

250 Supplement: The Theory of Assemblages.

algebraic point except the one x = 0, y = 1 . Of all the other number
pairs x, y which satisfy y = e? t one, at least, is transcendental. This
course of the exponential curve is certainly a most remarkable fact.

The full significance of these theorems which
reveal the existence in great quantity of numbers
which are not only not rational but which cannot
be represented by algebraic operations upon whole
numbers their significance for our concept of
the number continuum is tremendous. What
would Pythagorus have sacrificed after such a
discovery if the irrational seemed to him to merit
a hecatomb !

It is remarkable how little in general these
questions of transcendence are grasped and assim-
lg< " ilated, although they are so simple when one

has thought them through. I continually have the experience, in an
examination, that the candidate cannot even explain the notion "trans-
cendence". I often get the answer that a transcendental number satis-
fies no algebraic equation, which, of course, is entirely false, as the
example x e = shows. The essential thing, that the coefficients in
the equation must be rational, is overlooked.

If you will think our transcendence proofs through again you will
be able to grasp these simple elementary steps as a whole, and to make
them permanently your own. You need to impress upon your memory
only the Hermite integral; then everything develops itself naturally.
I should like to emphasize the fact that in these proofs we have used
the integral concept (or, speaking geometrically, the idea of area) as
something in its essence thoroughly elementary, and I believe that this
has contributed materially to the clearness of the proofs. Compare in
this respect, the presentation in Volume I of Weber- Wellstein, or in
my own little book, Vortrdge uber augewdhlte Fragen der Elementar-
geometrie 1 , where, as in the older school books, the integral sign is
avoided and its use replaced by approximate calculation of series
developments. I think that you will admit that the proofs there are
far less clear and easy to grasp.

These discussions concerning the distribution of the algebraic num-'
bers within the realm of real numbers lead us naturally to that second
modern field to which I have often referred during these lectures, and
which I shall now consider in some detail.

II. The Theory of Assemblages

The investigations of George Cantor, the founder of this theory, had
their beginning precisely in considerations concerning the existence of

1 Referred to p. 55. . '

The Power of an Assemblage. 251

transcendental numbers 1 . They permit one to view this matter in an
entirely new light. f

If the -brief survey of the theory of assemblages which I shall give
you has any special character, it is this, that I shall bring the treatment
of concrete examples more into the foreground than is usually done
in those very general abstract presentations which too often give this
subject a form that is hard to grasp and even discouraging.

1. The Power of an Assemblage

With this end in view, let me remind you that in our earlier dis-
cussions we have often had to do with different characteristic totalities
of numbers which we can now call assemblages of numbers. If I confine
myself to real numbers, these assemblages are

1. The positive integers.

2. The rational numbers.

3. The algebraic numbers.

4. All real numbers.

Each of these assemblages contains infinitely many numbers. Our
first question is whether, in spite of this, we cannot compare the magni-
tude or the range of these assemblages in a definite sense, i.e., whether
we cannot call the "infinity" of one greater than, equal to, or less than
that of another. It is the great achievement of Cantor to have cleared
up and answered this really quite indefinite question, by setting up
precise concepts. Above all we have to consider his concept of power
or cardinal number: Two assemblages have equal power (are equivalent)
when their elements can be put into one-to-one correspondence, i.e., when
the two assemblages can be so related to each other that to each element of
the one there correponds one element of the other, and conversely. If such
a mutual correspondence is not possible the two assemblages are of
different power \ if it turns out that, no matter how one tries to set up
a correspondence, there are always elements of one of the assemblages
left over, this one has the greater power.

Let us now apply this principle to the four examples given above.
It might* seem, at first, that the power of the positive integers would
"be smaller than that of the rational numbers, the power of these smaller
than that of the algebraic numbers, and this finally smaller than that
of all real numbers; for each of these assemblages arises from the pre-
ceding by the addition of new elements. But such a conclusion would
be too hasty. For although the power of a finite assemblage is always
greater than the power of a part of it, this theorem is by no means valid
for infinite assemblages. This discrepancy, after all, need not cause

1 See Journal fur Mathematik, vol. 77 (1873), p. 258.

252

Supplement: The Theory of Assemblages.

surprise, since we are concerned in the two cases with entirely different
fields. Let us examine a simple example which will show clearly that
an infinite assemblage and a part of it can actually have the same
power, the aggregate, namely, of all positive integers and that of all
positive even integers

1, 2, 3, 4, 5, 6, . . .,

2, 4, 6, 8, 10, 12,

The correspondence indicated by the double arrows is obviously of the
sort prescribed above, in that each element of one assemblage corresponds,
to one and only one of the other. Therefore, by Cantor's definition, the
assemblage of the positive integers and the partial assemblage of the
even integers have the same power.

You see that the question as to the powers of our four assemblages
is not so easily disposed of. The simple answer, which then appears
the more remarkable, consists in Cantor's great discovery of 1873'. The
three assemblages, the positive integers, the rational, and the algebraic
numbers, have the same power', but the assemblage of all real numbers has
another, namely, a larger power. An assemblage whose elements can be
put into one-to-one correspondence with the series of positive integers
(which has therefore the same power) is called denumerable. The above

theorem can therefore be stated as

Jl!!!!!!! follows : The assemblage of the rational

* i wi I M I * ! I ! I ! I .

as well as of the algebraic numbers is
denumerable', that of all real numbers
is not denumerable.

Let us first give the proof for ra-
tional numbers, which is no doubt
familiar to some of you. Every ra-
tional number (we shall include the ne-
gative ones) can be expressed unique-
ly in the form pjq, where p and q
are integers without a common divi-
sor, where, say, q is positive, while p'
may also be zero or negative. In
order to bring all these fractions p/q

into a single series, let us mark in a p q plane all points with integral
coordinates (p,q), so that they appear as points on a spiral path as
shown in Fig. 119. Then we can number all these pairs (p, q) so that
only one number will be assigned to each and all integers will be used
(see Fig. 119). Now delete from this succession all the pairs (p, q) which
do not satisfy the above prescription (p prime to q and q > 0) and number

. ^

I *

*'*

r-v

t
i ,

7-3

)

9 1

? *

C $ ;

<r mZ

^ H

^V<

"~i

W*
10-

*-H

-7

-2

I
|

<rv

ig. 1?9.

The Power of an Assemblage. 253

anew only those which remain (indicated in the figure by heavy points).
We get thus a series which begins as follows:

1 0-12 i -i -2 3 | f i . . . ,

one in which a positive integer is assigned to each rational number and
a rational number to each positive integer. This shows that the rational
numbers are denumerable. This arrangement of the rational numbers

Rational number: -2 -4 -f -f ^ 7 J 2. 3

^ \ T I I I I i i i i i i i L

Positive integer: 7 7V J 73 6 72 Z 11 5 10 7 3 6

Fig. 120.

into a denumerable series requires, of course, a complete dislocation of
their rank as to size, as is indicated in Fig. 120, where the rational
points, laid off on the axis of abscissas, are marked with the order of
their appearance in the artificial series.

We come, secondly, to the algebraic numbers. I shall confine myself
here to. real numbers, although the inclusion of complex numbers would
not make the discussion essentially more difficult. Every real algebraic
number satisfies a real integral equation

which we shall assume to be irreducible, i.e., we shall omit any rational
factors of the left-hand member, and also any common divisors of
a l9 a lt . . ., a n . We assume also that a is always positive. Then, as
is well known, every algebraic co satisfies but one irreducible equation
with integral coefficients, in this normal form; and conversely, every
such equation has as roots at most n real algebraic numbers, but perhaps
fewer, or none at all. If, now, we could bring all these algebraic equations
into a denumerable series we could obviously infer that their roots and
hence all real algebraic numbers are denumerable.

Cantor succeeded in doint this by assigning to each equation a
definite number, its index,

and by separating all such equations into a denumerable succession
'of classes, according as the index N = \ , 2, 3 , . . . In no one of these
equations can either the degree n or the absolute value of any coefficient
exceed the finite limit N, so that, in every class, there can be only a
finite number of equations, and hence, in particular, only a finite number
of irreducible equations. One can easily determine the coefficients by
trying out all possible solutions for a given N and can, in fact, write down
at once the beginning of the series of equations for small values of N .
Now let us consider that, for each value of the index N , the real
roots of the finite number of corresponding irreducible equations have

254 Supplement: The Theory of Assemblages.

been determined, and arranged according to size. Take first the roots,
thus ordered, belonging to index one, then those belonging to index
two, and so on, and number them in that order. In this way we shall
have shown, in fact, that the assemblage of real algebraic numbers is de-
numerable, for we come in this way to every real algebraic number
and, on the other hand, we use all the positive integers. In fact one
could, with sufficient patience, determine say the 75 63-rd algebraic
number of the scheme, or the position of a given algebraic number,
however complicated.

Here, again, our "denumeration" disturbs completely the natural
order of the algebraic numbers, although that order is preserved among
the numbers of like index. For example, two algebraic numbers so
nearly equal as 2/5 and 2001/5000 have the widely separated indices 7
and 7001 respectively; whereas ]/ 5, as root of x 2 - 5 = 0, has the
same index, 7, as 2/5.

Before we go over to the last example, I should like to give you
an auxiliary theorem which will supply us with another denumerable
assemblage, as well as with a method of proof that will be useful to us
later on. If we have two denumerable assemblages

a l9 a 2 , <z 3 , . . . and 6 lf ft 2 , 6 3 , . . . ,

then the assemblage of all a and all b which arises by combining these
two is also denumerable. For one can write this assemblage as follows:

1, &1 1 #2> &2> #3 6 3> *

and we can at once set this into a one-to-one relation with the series of
positive integers. Similarly, if we combine 3 , 4, . . . , or any finite number
of denumerable assemblages, we obtain likewise a denumerable assemblage .
But it does not seem quite so obvious, and this is to be our auxiliary
theorem, that the combination of a denumerable infinity of denumerable
assemblages yields also a denumerable assemblage. To prove this, let us
denote the elements of the first assemblage by a lf a 2 , a 3 , . . ., those of
the second by b l , 6 2 , 6 3 , . . . , those of the third by c l , c 2 , c 3 , . . . , and
so on, and let us imagine these assemblages written under one another.
Then we need only choose the elements of this totality according to
successive diagonals, as indicated in the following scheme:

The Power of an Assemblage. 255

The resulting arrangement

1 2 3 4 5 6 7 8 9 10 11 ...
a^ a 2 6j 3 6 2 C] a A 6 3 c 2 d x 5 . . .

reaches ultimately every one of the numbers a , 6 , c , . . . and brings it
into correspondence with a definite positive integer, which proves the
theorem. In view of this scheme one could call the process a "counting
by diagonals'*.

The large variety of denumerable assemblages which has thus been
brought to our knowledge might incline us to the belief that all infinite
assemblages are denumerable. To show that this is not true we shall
prove the second part of Cantor's theorem, namely, that the continuum of
all real numbers is certainly not denumerable. We shall denote it by (5^ be-
cause we shall have occasion later to speak of multi-dimensional continua.

(&! is defined as the totality of all finite real values x, where we
may think of x as an abscissa on a horizontal axis. We shall first show
that the assemblage of all inner points on the unit segment < x < 1
has the same power as (^ . If we represent the first assemblage on the
x axis and the second on the y axis, at right angles to it, then a one-to-one
correspondence between them will be established by a rising monotone
curve of the sort sketched in Fig. 121 (e.g., a branch of the curve
y = (\lri) tan" 1 x} . It is permissible, therefore, to think of the
assemblage of all real numbers between and 1 as standing for g t and
we shall do this from now on.

The proof by which I shall show you that x is not denumerable is
the one which Cantor gave in 1891 at the meeting of the natural scientists
in Halle. It is clearer and more susceptible of generalization than the
one which he published in 1873. The essential thing in it is the so-called
"diagonal process*', by which a real number is disclosed that cannot
possibly be contained in any assumed denumerable arrangement of all
real numbers. This is a contradiction, and (5^ cannot, therefore, be
denumerable.

We write all our numbers < x < 1 as decimals and think of them
as forming a denumerable sequence

= 0, a a a

= 0, b l ""

= 0,

where a, b, c are the digits 0, 1 , . . ., 9 in every possible choice and
arrangement. Now we must not forget that our decimal notation is

256 Supplement: The Theory of Assemblages.

not uniquely definite. In fact according to our definition of equality

we have 0.999 . . . = 1 .000 . . . , and we could write every terminating

decimal as a non-terminating one in which all the digits, after a certain

O 7 one, would be nines. This is one of the

first assumptions in calculating with
decimals (see p. 34). In order, then,
to have a unique notation, let us
assume that we are employing only in-
finite, non-terminating decimals; that
all terminating ones have been con-
^ x verted into such as have an endless
succession of nines; and that only in-
finite decimals appear in our scheme

Fig. 121. rr

above.

In order now to write down a decimal x which shall be different
from every real number in the table, we fix our attention on the digits
i, & >f c s , . . . of the diagonal of the table (hence the name of the pro-
cess). For the first decimal place of x' we select a digit a\ different
from a x ; for the second place a digit b' 2 different from 6 2 ; for the third
place a digit c' 3 different from c 3 ; and so on:

These conditions for a{ , b' 2 , c' 3 , . . . allow sufficient freedom to insure
that x 1 is actually a proper decimal fraction, not, e.g., 0.999 . . . = 1 ,
and that it shall not terminate after a finite number of digits; in fact,
we can select a\, b' 2 , c'%, . . . always different from 9 and 0. The x' is
certainly different from x since they are unlike in the first decimal
place, and two infinite decimals can be equal only when they coincide
in every decimal place. Similarly x' ^ x 2 , on account of the second
place; x' =j= #s> because of the third place; etc. That is, x', a proper
decimal fraction, is different from all the numbers x lf x 2 , # 3 , . . . of the
denumerable tabulation. Thus the promised contradiction has appeared
and we have proved that the continuum (^ is not denumerable.

This theorem assures us, a priori, the existence of transcendental
numbers; for the totality of algebraic numbers was denumetable and
could not therefore exhaust the non-denumerable continuum of all real
numbers. But, whereas all the earlier discussions exhibited only a
denumerable infinity of transcendental numbers, it follows here that
the power of this assemblage is actually greater, so that it is only now
that we get the correct general view. To be sure, those special examples
were of service in giving life to an otherwise somewhat abstract picture 1 .

[ l The existence of transcendental numbers was first proved by Liouville; in
an article which appeared in 1851 in vol. 16, series 1, of the Journal des math^mati-
ques, he gave an elementary method for constructing such numbers.]

The Power of an Assemblage. 257

Now that we have disposed of the one dimensional continuum it is
very natural to inquire about the two-dimensional continuum. Every-
body had supposed that there were more points in the plane than in the
straight line, and it attracted much attention when Cantor showed 1 that
the power of the two dimensional continuum ( 2 was exactly the same as
that of the one dimensional g^. Let us take for @ 2 the square with side
of unit length, and for (^ the unit segment (see Fig. 122). We shall
show that the points of these two aggregates Q ^ i

can be put into a one-to-one relation. The fact i 1 >*

that this statement seems so paradoxical de- y
pends probably on our difficulty in freeing our
mental picture of a certain continuity in the
correspondence. But the relation which we shall
establish will be as discontinuous or, if you
please, as inorganic as it is possible to be. It

will disturb everything which one thinks of as F f g 122

characteristic for the plane and the linear mani-
fold as such, with the exception of the "power 1 '. It will be as though
one put all the points of the square into a sack and shook them up
thoroughly.

The assemblage of the points of the square coincides with that of
all pairs of decimal fractions

x = o. a^a^a^ . . . , y = 0. b^b^ >

all of which we shall suppose to be non-terminating. We exclude points
on the boundary for which one of the coordinates (x, y) vanishes, i.e.,
we exclude the two sides which meet at the origin, but we include the
other two sides. It is easy to show that this has no effect on the power.
The fundamental idea of the Cantor proof is to combine these two
decimal fractions into a new decimal fraction z from which one can
obtain (x, y) again uniquely and which will take just once all the values
< z ^ \ when the point (x, y) traverses the square once. If we then
think of z as an abscissa, we have the desired one-to-one correpondence
between the square ( 2 and the segment (5^, whereby the agreement
concerning the square carries with it the inclusion of the end z = 1
of the segment.

One might try to effect this combination by setting

from which one could in fact determine (x,y) uniquely by selecting
the odd and even decimal places respectively. But there is an objection
to this, due to the ambiguous notation for decimal fractions. This z,
namely, would not traverse the whole of ^ when we chose for (x, y)

i Journal fur Mathematik, vol. 84 (1878), p. 242 et seq.

Klein, Elementary Mathematics. 17

258 Supplement: The Theory of Assemblages.

all possible pairs of non-terminating decimals, that is, when we traversed
all the points of E 2 . For, although z is, to be sure, always non-termi-
nating, there can be non-terminating values of z, such as

z = 0. CiC 2 c 4 C Q c s . . /,

which correspond only to a terminating x or y y in the present case to

the values

x = 0. c x OOO . . . , y = 0. C 2 c 4 c 6 c 8 . . .

This difficulty is best overcome by means of a device suggested by
J. Konig of Budapest. He thinks of the a,b,c not as individual digits
but as complexes of digits one might call them "molecules" of the
decimal fraction. A "molecule" consists of a single digit, different from
zero, together with all the zeros which immediately precede it. Thus
every non-terminating decimal must contain an infinity of molecules,
since digits different from zero must always recur; and conversely. As
an example, in

x = 0.320 8007 000 302 405 ...

we should take as molecules a^ = 3, a z = 2, a 3 08, 4 = 007, # 6
= 0003 , a 6 = 02, 7 = 4, . . .

Now let us suppose, in the above rule for the relation between x, y
and z, that the a,b,c stand for such molecules. Then there will corres-
pond uniquely to every pair (x, y) a non-terminating z which would,
in its turn, determine x and y. But now every z breaks up into an x
and a y each with an infinity of molecules, and each z appears therefore
just once when (x, y) run through all possible pairs of non terminating
decimal fractions. This means, however, that the unit segment and
the square have been put into one-to-one correspondence, i.e., they
have the same power.

In an analogous way, of course, it can be shown that the continuum
of 3 , 4 , ... dimensions has the same power as the one dimensional
segment. It is more remarkable, however, that the continuum (S^, of
iiifinitely many dimensions, or more exactly, of denumerably infinitely
many dimensions, has also the same power. This infinite dimensional
space is defined as the totality of the systems of values whi^h can be
assumed by the denumerable infinity of variables

when each, independently of the others, takes on all real values. This
is really only a new form of expression for a concept that has long been
in use in mathematics. When we talk of the totality of all power series
or of all trigonometric series, we have, in the denumerable infinity of
coefficients, really nothing but so many independent variables which,
to be sure, are for purposes of calculation restricted by certain require-
ments to ensure convergence.

The Power of an Assemblage. 259

Let us again confine ourselves to the "unit cube" of the (S^, i.e.,
to the totality of points which are subject to the condition < x n ^ 1 ,
and show that they can be put into one-to-one relation with the points
of the unit segment < z ^ 1 of S^. For convenience, we exclude all
boundary points for which one of the coordinates x m vanishes, as well
as the end point z = 0, but admit the others. As before we start with
the decimal fractional representation of the coordinates in K^:

*, = 0, ^&i <* 2 3

x 2 = O t b l b 2 b 3 .

*3 = 0, Ct C 2 C 3

where we assume that the decimal fractions are all written in non-
terminating form, and furthermore that the a, b, c y . . . are "decimal
fraction molecules 1 ' in the sense indicated above, i.e., digit complexes
which end with a digit which is different from zero, but which is preceded
exclusively by zeros. Now we must combine all these infinitely many
decimal fractions into a new one which will permit recognition of its
components; or, if we keep to the chemical figure, we wish to form such
a loose alloy of these molecular aggregates that we can easily separate
out the components. This is possible by means of the "diagonal process"
which we applied before (p. 254). From the above table we get, according
to that plan

z = 0, a a 2 b a< 3 b 2 c^ a b Q c 2 d l a 5 . . . ,

which relates uniquely a point of x to each point of (00. Conversely
we get in this way every point z of K lf for from the non terminating
decimal fraction for a given z we can derive, according to the above
given scheme, an infinity of non-terminating decimals x l , x 2 , x 3 , . . . ,
out of which this z would arise by the method indicated. We have
succeeded therefore in setting up a one-to-one correspondence between
the unit cube in (5^ and the unit segment in IB

Our results thus far show that there are at least two different
powers :

1. That of the denumerable assemblages.

2. That of all continua ( 1 ,( 2 >@'3> . . ., including (S^.

The question naturally arises whether there are still larger powers.
The answer is that one can exhibit an assemblage having a still higher
power, not merely as a result of abstract reasoning, but one lying quite
within the range of concepts which have long been used in mathematics.
This aggregate is, namely:

3. That of all possible real functions / (x) of a real variable x.

17*

260 Supplement: The Theory of Assemblages.

It will be sufficient for our purpose to restrict the variable to the
interval < x < \ . It is natural to think first of the aggregate of the
continuous functions / (x) , but there is a remarkable theorem which
states that the totality of all continuous functions has the same power
as the continuum, and belongs therefore in group 2. We can reach a
new, a higher power, only by admitting discontinuous functions of
the most general kind imaginable, i.e., where the function value at any
place x is entirely arbitrary and has no relation to neighboring values.

I shall first prove the theorem concerning the aggregate of continuous
functions. This will involve a repetition and a refinement of the con-
siderations which we adduced (p. 206) in order to make plausible the
possibility of developing "arbitrary" functions into trigonometric series.
At that time I remarked:

a) A continuous function / (x) is determined if one knows the values
/ (r) at all rational values of r .

b) We know now that all rational values r can be brought into a
denumerable series r l9 r 2 , ? 3 , . . .

c) Consequently f(x) is determined when
one knows the denumerable infinity of quan-
tities /(r x ), f(r 2 ), /(r s ), . . . Moreover, these
values cannot, of course, be assumed arbit-
rarily if we are to have a single-valued con-
tinuous function. The assemblage then of all
possible systems of values / (r^ , / (r 2 ) , . . .
+x must contain a sub-assemblage whose power
is the same as that of the assemblage of all

Fig. 123 G

continuous functions (see Fig. 123).

d) Now the magnitudes / x = / (r^ , f 2 = / (r 2 ) , . . . can be considered
as the coordinates of a (00, since they make up a denumerable infinity
of continuously varying magnitudes. Hence, in view of the theorem
already proved, the totality of all their possible systems of values has
the power of the continuum.

e) Since the assemblage of continuous functions is contained in an
assemblage which is equivalent to the continuum, it must itself be
equivalent to a sub-assemblage of the continuum.

f) But it is not hard to see that, conversely, the entire continuum
can be put into one-to-one correspondence with a part of the assemblage
of all continuous functions. For this purpose, we need to consider only
the functions defined by / (x) = k = const., where A; is a real parameter.
If k traverses the continuum j then / (x) will describe an assemblage
which is in one-to-one correspondence with (^ but which is only a part
of the totality of all continuous functions.

g) Now we must make use of an important general theorem of the
theory of assemblages, the so-called theorem of equivalence, due to

The Power of an Assemblage. 261

F. Bernstein 1 : // each of two assemblages is equivalent to a part of the
other then the two assemblages are equivalent. This theorem is very plau-
sible. The proof of it would take us too far afield.

h) According to e) and f) the continuum (5^ and the aggregate of
all continuous functions satisfy the conditions of the theorem of equi-
valence. They are therefore of like power, and our theorem is proved.

Let us now go over to the proof of our first theorem, that the as-
semblage of all possible functions that are really entirely arbitrary has
a power higher than that of the continuum. The proof is an immediate
application of Cantor's diagonal process.

a) Assume the theorem to be false, i.e., that the assemblage of all
functions can be put into one-to-one correspondence with the conti-
nuum (]_. Suppose now, in this one-to-one relation, that the function
/ (x, v) of % corresponds to the value x = v in lf so that, while v tra-
verses the continuum (^ , / (x , v) represents all possible functions of x .
We shall reduce this supposition to an absurdity by actually setting
up a function F (x) which is different from all such functions / (x, v).

b) For this purpose we construct the "diagonal function" of the
tabulation of the f(x,v), i.e., that function which, for every value
x = X Q , has that value which the assumed correspondence imposes upon
/ (x, v) when the parameter v also has the value v = X Q , namely / (x , x ).
Written as a function of x , this is simply the function / (x , x) .

c) Now we construct a function .-F (x) which for every x is different

from this f(x, x):

F(x) 4= f(x, x) for every x.

We can do this in the greatest variety of ways, since we admit the most
completely discontinuous functions, whose value at any place can be
arbitrarily determined. We might, for example, put

F(x)=f(x,x) + i.

d) This F(x) is actually different from every one of the functions
f(x,v). For, if F(x) = f(x, v ) for some v = r Qt the equality would
hold also for x = v ; that is, we should have F(r Q ) = /(>' >'o), which
contradicts the assumption in c) concerning F (x) .

The assumption a) that the functions f(x,v) could exhaust all func-
'tions is thus overthrown, and our theorem is proved.

It is interesting to compare this proof with the analogous one for
the non-denumerability of the continuum. There we assumed the
totality of decimal fractions arranged in a denumerable table; here we
consider the function scheme f(x,v). The singling out there of the
diagonal elements corresponds to the construction here of the diagonal
function f(x, x) ; and in both cases the application was the same, namely

1 First published in Borel's Lemons sur la Th&oiie des Fonctions, Paris, 1898,
p. 103-

262 Supplement: The Theory of Assemblages.

the setting up of something new, i.e., not contained in the table, in
the one case a decimal fraction, in the other a function.

You can readily imagine that similar considerations could lead us
to assemblages of ever increasing power beyond the three which we
have already discussed. The most noteworthy thing in all these results
is that there remain any abiding distinctions and gradations at all in
the different infinite assemblages, notwithstanding our having subjected
them to the most drastic treatment imaginable; treatment which
disturbed special properties, such as order, and permitted only the
ultimate elements, the atoms, to retain an independent existence as
things which could be tossed about in the most arbitrary manner. And
it is worth noting that the three gradations which we did establish were
among things which have long been familiar in mathematics integers,
continua, and functions.

With this I shall close this first part of my discussion of the theory
of assemblages, which has been devoted mainly to the concept of power.
In a similar concrete manner, but with still greater brevity, I shall now
tell you something about a farther chapter of this theory.

2. Arrangement of the Elements of an Assemblage
We shall now bring to the front just that thing which we have
heretofore purposely neglected, the question, namely, how individual
assemblages of the same power differ from one another by virtue of
those relations as to the arrangement of the elements which are intrinsic
in the assemblage. The most general one-to-one representations which
we have admitted thus far disturbed all these relations, think only
of the representation of the square upon the segment. I desire to
emphasize, especially, the significance of precisely this chapter of the
theory of assemblages. It cannot possibly be the purpose of the theory
of assemblages to banish the differences which have long been so familiar
in mathematics, by introducing new concepts of a most general kind.
On the contrary, this theory can and should aid us to understand those
differences in their deepest essence, by exhibiting their properties in
new light.

We shall try to make clear the different possible arrangements, by
considering definite familiar examples. Beginning with denumerable
assemblages, we note three examples of fundamentally different ar-
rangement, so different that the equivalence of their powers was, as
we saw, the result of a special and by no means obvious theorem. These
examples are:

1. The assemblage of all positive integers.

2. The assemblage of all (negative and positive) integers.

3. The assemblage of all rational numbers and that of all algebraic
numbers.

Arrangement of the Elements of an Assemblage. 263

All these assemblages have a common property in the arrange-
ment of their elements, which finds expression in the designation
simply ordered, i. e., of two given elements, it is always known
which precedes the other, or, put algebraically, which is the smaller
and which the larger. Further, if three elements a, b, c are given,
then, if a precedes b and b precedes c, a precedes c (if a < b and b < c
then a <c).

But now as to the characteristic differences. In (1), there is a first
element (one) which preceded all the others, but no last which follows
all the others; in (2), there is neither a first nor a last element. Both
(1) and (2) have this in common, that every element is followed by
another definite one, and also that every element [except the first in
(1)] is preceded by another definite one. In contrast with this, we find
in (3) (as we saw p. 31) that between any two elements there are always
infinitely many others the elements are "everywhere dense", so that
among the rational or the algebraic numbers lying between a and b
there is neither a smallest nor a largest. The manner of arrangement
in these three examples, the type of arrangement (Cantor's term type
of order seems to me less expressive) is quite different, although the
power is the same. One could raise the question here as to all the types
of arrangement that are possible in denumerable assemblages, and that
is what the students of the theory of assemblages actually do.

Let us now consider assemblages having the power of the continuum.
In the continuum x of all real numbers, we have a simply ordered
assemblage; but in the multidimensional types ( 2 , 3 , . . . we have
examples of an order no longer simple. In the case of S 2 , for instance,
two relations are necessary, instead of one, to determine the mutual
position of two points.

The most important thing here is to analyze the concept of continuity
for the one dimensional continuum. The recognition of the fact that
continuity here depends on simple properties of the arrangement which
is peculiar to C l , is the first great achievement of the theory of as-
semblages toward the clarifying of traditional mathematical concepts.
It was found, namely, that all the continuity properties of the ordinary
continuum flow from its being a simply ordered assemblage with the
following two properties:

1 . If we separate the assemblage into two parts A , B such that every
element belongs to one of the two parts and all the elements of A precede
all those of B , then either A has a last element or B a first element. If we
recall Dedekind's definition of irrational number (see p. 33 e * sec l-)
we can express this by saying that every "cut" in our assemblage is
produced by an actual element of the assemblage.

2. Between any two elements of the assemblage there are always in-
finitely many others.

264 Supplement: The Theory of Assemblages.

Thi.s second property is common to the continuum and the de-
numerable assemblage of all rational numbers. It is the first property
however that marks the distinction between the two. In the theory
of assemblages it is customary to call all simply-ordered assemblages
continuous if they possess the two preceding properties, for it is actually
possible to prove for them all the thorems which hold for the continuum
by virtue of its continuity.

Let me remind you that these properties of continuity can be
formulated somewhat 'differently in terms of Cantor's fundamental
series. A fundamental series is a simply-ordered denumerable series of
elements a lt a 2 , a 3 , . . . of an aggregate such that each element of the
series precedes the following or each succeeds it:

a l < a 2 < a 3 < . . . or a l > a 2 > a 3 > . . .

An element a of the aggregate is called a limit element of the fundamental
series if (in the first sort) every element which precedes a but no element
which follows a is ultimately passed by elements of the fundamental
series; and similarly for the second sort. Now if every fundamental
series in an aggregate has a limit element, the aggregate is called closed ;
if, conversely, every element of the aggregate is a limit element of a
fundamental series, the aggregate is said to be dense. Now continuity,
in the case of aggregates having the power of the continuum, consists
essentially in the union of these two properties.

Let me remind you incidentally that when we were discussing the
foundations of the calculus we spoke also of another continuum, the
continuum of Veronese, which arose from the usual one by the addition
of actually infinitely small quantities. This continuum constitutes a
simply-ordered assemblage in as much as the succession of any two
elements is determinate, but it has a type of arrangement entirely
different from that of the customary S^; even the theorem that every
fundamental series has a limit element no longer holds in it.

We come now to the important question as to what representations
preserve the distinctions among the continua Si,S 2 , of ^different
dimensions. We know, indeed, that the most general one-to-one re-
presentation obliterates every distinction. We have here the important *
theorem that the dimension of the continuum is invariant with respect
to every continuous one-to-one representation, i.e., that it is impossible
to effect a reversibly unique and continuous mapping of a ( m upon
a ( n where m =j= n. One might be inclined to accept this theorem,
without further ado, as self evident ; but we must recall that our naive
intuition seemed to exclude the possibility of a reversibly unique
mapping of ( 2 upon (5^ , and this should dispose us to caution in accepting
its pronouncements.

Arrangement of the Elements of an Assemblage. 265

I shall discuss in detail only the simplest case 1 , which concerns the
relation between the one-dimensional and the two-dimensional continua,
and I shall then indicate the difficulties in the way of an extension to
the most general case. We shall prove, then, that a reversibly unique,
continuous relation between ^ and ( 2 is not possible. Every word here
is essential. We have seen, indeed, that we may not omit continuity;
and that reversible uniqueness may not be omitted is shown by the
example of the "Peano curve" which is doubtless familiar to some of you.

We shall need the following auxiliary theorem: Given two one-
dimensional continua (^ , (&i which are mapped continuously upon each
other so that to every element of &{ there corresponds one and but one element
of C lf and to every element of C x there corresponds at most one element of
(/; if, then, a, b are two elements of x to which two elements a' t b'
in (&! actually correspond, respectively, it follows that to every element c
of &! lying between a and b there

will correspond an element c 1 of _i _ , - 1 -- , - 1 . - jc t
i which lies between a 1 and V *? _ f ? _

(see Fig. 124). This is analogous Fig. 124.

to the familiar theorem that a

continuous function f(x) which takes two values a, b at the values

% = a 7 , V must take a value c , chosen arbitrarily between a and 6, at

some value c' between a' and V \ and it could be proved as an exact

generalization of this theorem, by using the

above definition of continuity. This would J - ' - ' -^

require one also to explain continuous map-

ping of a continuous assemblage in a manner

analogous to the usual definition of continu-

ous functions, and it can be done with the

aid of the concept of arrangement. But this

is not the place to amplify these ideas. Fig. 125.

We shall give our proof as follows. We

assume that a continuous reversibly unique mapping of the one di-
mensional segment Kj upon the square ( 2 has been effected (see
Fig. 125). Let two elements a, b on (5^ correspond to the elements
A, B, respectively, of 2 . Now we can join these elements A,B by
two different paths within ( 2 , e.g., by the broken lines i,(i drawn
in the figure. To do this, it is not necessary to presuppose any
special properties of 2 , such as the setting up of a coordinate system;
we need merely use the concept of double order. Each of the paths
i and Si will be a simply-ordered one-dimensional continuum like (1,
and because of the continuous reversibly unique relation between ( r
and ( 2 there must correspond just one point on E t to each element of

1 Brouwer, L. E. J. gave a proof for the general case in 1911, in volume 70,
p. 161, of the Mathematische Annalen.

266 Supplement: The Theory of Assemblages.

(5 and &{ ; but to each element of S x there must correspond at most one
on i or (i . In other words, we have precisely the conditions of the
above lemma, and it follows that to every point c in 6^ between a and
b there corresponds not only a point c' of 6^ but also a point ~' of &i .
But this contradicts the assumed reversible uniqueness of the relation
between (^ and 2 . Consequently this mapping is not possible and the
theorem is proved.

If one wished to extend these considerations to two arbitrary
continua ( w , n> it would be necessary to know in advance something
about the constitution of continua of general nature and of dimension
1 , 2, 3 > - , w 1 , which can be embedded in & m . As soon as m y
n^2 one can not get along merely with the concept "between' as
we could in the simplest case above. On the contrary, one is led to very
difficult investigations which include, among the earliest cases, the
abstruse fundamental geometric questions concerning the most general
continuous one-dimensional assemblage of points in the plane, questions
which only recently have been somewhat cleared up. One of these
interesting questions is as to when such an assemblage of points should
be called a curve.

I shall close with this my very special discussion of the theory of
assemblages, in order to add a few remarks of a general nature. First,
a word as to the general notions which Cantor had entertained concerning
the position of the theory of assemblages with reference to geometry
and analysis. These notions exhibit the theory of assemblages in a
special light. The difference between the discrete magnitudes of arith-
metic and the continuous magnitudes of geometry has always had a
prominent place in history and in philosophical speculations. In recent
times the discrete magnitude, as conceptually the simplest, has come
into the foreground. According to this tendency we look upon natural
numbers, integers, as the simplest given concepts ; we derive from them
in the familar way, rational and irrational numbers, and we construct
the complete apparatus for the control of geometry by means of analysis,
namely, analytic geometry. This tendency of modern development
can be called that of arithrrietizing geometry. The geometric 'idea of
continuity is carried back to the idea of whole numbers. These lectures
have, in the main, held to this direction.

Now, as opposed to this one-sided preference for integers, Cantor
would (as he himself told me in 1903 at the meeting of the natural
scientists in Cassel) achieve, in the theory of assemblages, "the genuine
fusion of arithmetic and geometry". Thus the theory of integers, on
one hand, as well as the theory of different point continua, on the other,
and much more, would form a homogeneous group of equally important
chapters in a general theory of assemblages.

Arrangement of the Elements of an Assemblage. 267

I shall add a few general remarks concerning the relation of the theory
of assemblages to geometry. In our discussion of assemblages we have
considered:

1 . The power of an assemblage as something that is unchanged by
any reversibly unique mapping.

2. Types of order of assemblages which take account of 'the relations
among the elements as to order. We were able here to characterize the
notion of continuity, the different multiple arrangements or multi-
dimensional continua, etc., so that the invariants of continuous map-
pings found their place here. When carried over to geometry, this gives
the branch which, since Riemann, has been called analysis situs, that most
abstract chapter of geometry, which treats those properties of geometric
configurations which are invariant under the most general reversibly
unique continuous mappings. Riemann had used the word manifold
(Mannigfaltigkeit) in a very general sense. Cantor used it also, at first,
but replaced it later by the more convenient word assemblage (Menge).

3. If we go over to concrete geometry we come to such differences
as that between metric and projective geometry. It is not enough here
to know, say, that the straight line is one-dimensional and the plane
two-dimensional. We desire rather to construct or to compare figures,
for which we need to use a fixed unit of measure or at least to choose
a line in the plane, or a plane in space. In each of these concrete domains
it is necessary, of course, to add a special set of axioms to the general
properties of arrangement. This implies, of course, a further develop-
ment of the theory of simply-ordered, doubly-ordered, . . ., n-tuply-
ordered, continuous assemblages.

This is not the place for me to go into these things in detail,
especially since they must be taken up anyway in the succeeding vo-
lumes of the present work. I shall merely mention literature in which
you can inform yourselves farther. Here, above all, I should speak of
the reports in the Mathematische Enzyklopadic : Enriques, Prinzipien
der Geometrie (III. A. B. 1) and v. Mangoldt, Die Begriffe ,,Linie tf und
,,Flache" (III. A. B. 2), which treat mainly the subject of axioms; also
Dehn-Heegaard, Analysis situs (III. A. B. 3). The last article is written
in rather abstract form. It begins with the most general formulation
of the concepts and fundamental facts of analysis situs, as these were
set up by Dehn himself, from which everything else is deduced then
by pure logic. This is in direct opposition to the inductive method of
presentation, which I always recommend. The article can be fully
understood only by an advanced reader who has already thoroughly
worked the subject through inductively.

As to literature concerning the theory of aggregates, I should men-
tion, first of all, the report made by A. Schoenflies to the Deutsche
Mathematikervereinigung, entitled: Die Entwickelung der Lehre von

268 Supplement: The Theory of Assemblages.

den Punktmannigfaltigkeiten 1 . The first part appeared in volume 8 'of
the Jahresbericht der deutschen Mathematikervereinigung; the second
appeared recently as a second supplementary volume to the Jahres-
bericht. This work is really a report on the entire theory of aggregates,
in which you will find information concerning numerous details. Along-
side of this, I would mention the first systematic textbook on the
theory of aggregates: The Theory of Sets of Points, by W. H. Young
and his wife, Grace Chisholm Young (whom we mentioned p. 179)-

In concluding this discussion of the theory of assemblages we must
again put the question which accompanies all of our lectures: How
much of this can one use in the schools? From the standpoint of mathe-
matical pedagogy, we must of course protest against putting such
abstract and difficult things before the pupils too early. In order to
give precise expression to my own view on this point, I should like to
bring forward the biogenetic fundamental law, according to which the
individual in his development goes through, in an abridged series, all
the stages in the development of the species. Such thoughts have become
today part and parcel of the general culture of everybody. Now, I think
that instruction in mathematics, as well as in everything else, should
follow this law, at least in general. Taking into account the native
ability of youth, instruction should guide it slowly to higher things,
and finally to abstract formulations; and in doing this it should follow
the same road along which the human race has striven from its naive
original state to higher forms of knowledge. It is necessary to formulate
this principle frequently, for there are always people who, after the
fashion of the mediaeval scholastics, begin their instruction with the
most general ideas, defending this method as the "only scientific one".
And yet this justification is based on anything but truth. To instruct
scientifically can only mean to induce the person to think scientifically,
but by no means to confront him, from the beginning, with cold, sci-
entifically polished systematics.

An essential obstacle to the spreading of such a natural and truly
scientific method of instruction is the lack of historical knowledge which
so often makes itself felt. In order to combat this, I have made a point
of introducing historical remarks into my presentation. By doing this
I trust I have made it clear to you how slowly all mathematical ideas
have come into being; how they have nearly always appeared first in
rather prophetic form, and only after long development have crystallized
into the rigid form so familiar in systematic presentation. It is my
earnest hope that this knowledge may exert a lasting influence upon
the character of your own teaching.

1 2 parts, Leipzig 1900 and 1908, A revision of the first half appeared in 1913
under the title: Entwickelung dev Mengenlehre und ihrer Anwendungen; as a continu-
ation of this, see H. Hahn: Theorie der reellen Funktionen, vol. I, Berlin, 1921.

Index of Names.

Abel 84, 138, 154.
d'Alembert 103, 212.
Archimedes 80, 209, 219,
222, 237-

Bachmann 39, 48.
Ball 74.
Baltzer 72.
Bauer 86.
Baumann 220.
Berkeley 219-
Bernoulli, Daniel 205-
, Jacob 200.
, Johann 200, 205, 216.
Bernstein 261.
Bessel 191.
Braunmuhl 175-
Briggs 172, 173.
Brouwer 265-
Budan 94.
Biirgi 147-
Burkhardt23,29, 191,205-

Cantor, Georg 12, 32, 35,
204, 221, 250, 266,267-

Cardanus 55, 80, 134.

Cartesius, see Descartes.

Cauchy 84, 154, 202, 213,
219, 228, 231, 235-

Cavalieri 210, 214.

Cayley 6j8, 73, 74.

Chernac 40.

Chisholm 179.

Clebsch 84.

Coble 143-
. Copernicus 8 1 , 171.

Coradi 198.

Dedekind 13, 33-
Dehn 267.
Delambre 180, 181.
De Moivre 153, 168.
Descartes 81, 94.

Dirichlet 42, 199, 202, 203,

204, 206-
Dyck 94.

Enriques 55, 267.
Eratosthenes 40.
Eudoxus 219.
Euklid 32, 80, 219-
Euler 50, 56, 77, 82, 155,

166, 200, 202, 212, 234,

237.

Fejer 200.
Fermat 39, 48, 58.
Fourier 91, 201, 204, 206.
207, 222, 236.

Galle 17.

Gauss 39, 42, 50, 58, 76,

102, 154, 181.
Gibbs 199, 200.
Gordan 143-
Goursat 235.
Grassmann 12, 58, 64.
Gutzmer 2.

Hahn 268.

Hamilton 1 1, 58, 62, 73, 74.

Hammer 175.

Hankel 26, 56.

Harnack 235-

Hartenstein 99.

Heegard 267-

Hegel 217.

Heiberg 80, 209-

Hermite 238, 239, 245-

Hilbert 13, 14, 48, 21 8,

238, 243-
1'Hospital 216.

Jacobi 84.

Kant 10.

Kitstner 76, 210, 212.
Kepler 208, 210.
Kimura 74.

Konig, J. 258.
Kowalewski 215, 216.
Kummer 48.

Lacroix 235-

Lagrange 66, 82, 83, 153,

200, 220, 222, 234.
Leibniz 13, 20, 56, 82, 200,

211, 214, 215, 220, 222.
Lie 84.

Lindemann 238, 243, 249-
Liouville 256.
Liibsen 216.
Liiroth 17.

Maclaurin 210, 212, 234-
Mannchen 49-
Mangold 267.
Markoff 236.
Mehmke 95, 170.
Mercator, N. 81, 150, 168.
Michelson 198, 199-
Minkowski 11, 39.
Mobius 176, 177, 182.
Molk, J. 8.
Mollweide l8l.
Monge 84.

Napier, see Neper.
Neper 81, 147,150, 172, 173-
Netto 86.
Newton 81, 82, 151, 168,

210, 212, 222, 230, 233,

237-
Norlund 236.

Odhner 17.
Ohm 76.
Ostrowski 103-

Peano 12, 265-
Peurbach 171.
Picard 84, 160.
Pitiscus 172, 174.

270

Index of Names.

Plato 80*120.

Poisson 216.
Pringsheim, A. 233-
Ptolemy 170.
Pythagorus 31, 250.

Regiomontanus 171.

Rhaticus 171.

Riemann 84, 159, 202,

267-
Runge 86, 92, 191. 198.

Schafheitlein 216.
Scheffers 155, 235-
Schellbach 222.

| bchimmack 3, 194,223, 224.
Schlomilch 235-
Schoenfliess 267.
Schubert 8.
Seeger 189.
Serret 86, 235-
Shanks 237-

Simon 5, 24, 85, 162, 221.
Stifel 146.
Stratton 198.
Study 175, 181.
Sturm, J. 94.

Tannery, J. 8.

Taylor 82, 153, 227, 232,

233, 234, 236.
Timerding 189.

Tropjke 28,* 85, 170.

Vega 173.

Veronese 218, 264.'
Vieta 25.
Vlacq 173- -

Weber 4, 13, 23, 29, 86,

175, 182, 250.
Weierstrass 33, 84, 202,

203, 213.
Wilbraham 198.
Wolff, Chr. 216.
Wolfskehl 48.
Wiillner 217.

Young, G. Chisholm 1 80.
-,W. H. 268.

Index of Contents.

Abridged reckoning 10 et seq.
Actually infinitely small quantities 214,

218, 219-
Algorithmic method, see Processes of

growth, plan C.
Analysis situs 267-

Applicability and logical consistency in
infinitesimal calculus 221.
in the theory of fractions 29-
complex numbers 56 58.
irrational numbers 33.
natural numbers 14.
negative numbers 23 25.
Applied mathematics 4, 15-
Approximation, mathematics of 36.
Archimedes, axiom of 218.
Arithmetization 266.
Arrangement within an assemblage 262.
Assemblage of continuous and real func-
tions 206, 259-261.

of algebraic and transcendental
numbers 250, 254 256.

Branch points 107, 109-

Calculating machines 17 21.

and formal rules of operation 21, 22.
Cardinal number 251-

Casus irreducibilis of the cubic equation

135-
Circular functions:

analogy with hyperbolic functions
166?

see also trigonometric functions.
Closed fundamental series 264.
Complex numbers, higher 5875.
Consistency, proofs of 13, 25, 57.

and applicability :

of infinitesimal calculus 221.
of the theory of fractions 30.

complex numbers 55 58.

irrational numbers 34.

natural numbers 14.

negative numbers 23.

Constructions with ruler and compasses

49-

Continued fractions 4244.
Continuity, analysis of, based on theory

of assemblages 263 266.
Curriculum proposals, the Meran 16.
Cut, after Dedekind 33.
Cyclometric functions:

definition of, by means of quadra-
ture of the circle 163 168. ,
Cyclotomic numbers 47-

Decimal system 6, 9, 20.
Dense 31, 249, 263, 264.
Dcnumerability of algebraic numbers
i 253 et seq.

rational numbers 252 et seq.

a denumerable infinity of de-
numerable assemblages 254.

Derivative calculus 220, 234.
Development of infinitesimal calculus

208-220.

Diagonal process 254, 259, 261.
Differences, calculus of 228, 230232.
Differentials, calculation with:

naive intuitional direction 208 210.

direction of mathematics of ap-
proximation 215, 216.

formal direction 215.

speculative direction 214, 216, 217.
Dimension, in variance of the of a

continuum by reversibly unique

mapping 264, 265-
Discriminant curve of the quadratic and

cubic equation 92.

surface of the biquadratic equation
98101.

Equations :

cyclotomic 50.

pure 110-115, 131-134.

reciprocal 51.

of fifth degree 141-142.

the dihedral 115 120, 126.

the tetrahedral 120130.

272

Index of Contents.

the octahedral 120130.
the icosahedral 120130.
Equivalence, of assemblages 251262.
, theorem of 260.
Exhaustion, method of 209.
Exponential function:

definition by quadrature of hyper-
bola 149 et seq., 156-157-
general , and e w 158 159,

160-161-
series for e x 152.

function -theoretic discussion of 156
et seq.

Fermat, great theorem of 4649-
Formal mathematics 24, 26, 29, 56.
Foundations of arithmetic:
by means of intuition 11.

formalism 13.

logic 11.

theory of point sets 12.
Fourier 's series, see trigonometric series.

integral 207-
Function, notion of:

analytic function 200201.

arbitrary function 200.

relation of the two in complex region
202-203-

discontinuous real functions 204.
Functions, assemblage of continuous

and real 206, 261262.
Fundamental laws of addition and

multiplication 810.

logical foundation 1016.

consistency 13 et seq.

regions on the sphere 111 114,
117-120.

series, Cantor's 264.

theorem of algebra 101 104.
Fractions, changing common into deci-
mal 40.

Gamma function 239.
Graphical methods for equations in the
complex field 102133.

determining the real solutions
of equations 87 101.

Historical excursus on :

relations between differential cal-
culus and the calculus of finite
differences 232235.

exponential function and logarithm
146-155-

the notion of function 200207.

infinitesimal calculus 207 223.

imaginary numbers 55, 75 76.

irrational numbers 31 34.

negative numbers 25 27.

Taylor's theorem 233 234-

transcendence of e and n 237 238.

trigonometric series 205207.

trigonometric tables and logarithmic
tables 170-174.

the modern development and the
general structure of mathematics

77-85.
Homogeneous variables in function

theory 106-108.
Hyperbolic functions 164166.

analogy with circular functions 166.

fundamental function for 166.

Impossibility, proofs of:

general 51-

construction of regular heptagon
with ruler and compasses 51 55-

trisection of an angle 114.
Induction, mathematical 11.
Infinitesimal calculus, invention and

development of 207 et seq.
Instruction, reform in 5.
Interpolation :

by means of polynomials after
Lagrange 229-
Newton 229232.

trigonometric 190193-
Interpolation parabolas 229-
Investigation, mathematical 208.
Irreducibility : ^

function-theoretic 113114.

number-theoretic 52.

Lagrange f s interpolation formula 229.
Limit, method of 211-214.
Logarithm :

base of the natural 150151-
calculation of 148 et seq., 1 72 et seq.
definition of the natural by means
of quadrature of the hyperbola
149, 156.

difference equation for the 148.
function -theoretic discussion of

156-162.

uniformization by means of 133,
159-

Mean-value theorem of differential
calculus 213-214; extension of
same 231 et seq.

Index of Contents.

273

Newton's interpolation formula 229

to 232.
Nomographic scales for:

order curves 89, 94.

class curves 90, 95-
Non-denumerability of the continuum

256.

Non-Archimedean number system 218.
Normal class curve of biquadratic

equation 9698.

curves as:

class curves 9093, 95, 97-
order curves 8990, 94.

equations of the regular bodies:
solution by separation and series

130-133-

uniformization 133 138.

- - radicals 138141.

reduction of general equations to

normal equations 141 143-
Number, assemblage of continuous and

real numbers 250, 251253-
, notion of 10.

, transition from, to measure 28.
- pair 28, 56.

scale 23, 26, 31.

Order, types of 263-
Osculating parabolas 224 226.
limiting form of 227.

Peano curve 265-
Perception, inner n.

and logic 11.
Philologists, relation to 2.
Picard's theorem 160.

Point, the infinitely distant of the

complex plane 105-
Point lattice 43-
Power of the continuum of a de-

numerable infinity of dimensions 258 .

of a finite number of dimensions
257-258.

of ar? assemblage 251262.

the assemblage of all real func-

tions 261.

continuous functions 260.
Precision, mathematics of 36.
Prime numbers, existence of infinitely

many 40.

factor tables 40.
Principle of permanence 26.
Process of growth of mathematics:

Plan A. Separating methods and

disciplines; logical direction 75.
Klein, Elementary Mathematics.

Plan B. Fusing methods and dis-
ciplines; intuitive direction 77.
Plan C. Algorithmic process; for-
mal direction 79.
Psychologic moments in teaching 4, 10,

16, 28, 30, 34, 268.
Pythagorean numbers 44.

Quaternion 6075-

scalar part of 60.

vector part of 60.

tensor of 63, 66, 72.

versor of 72.

Rational, in the sense of mathematics of

approximation 36.
Reform, the Basel aims toward 2.

movement:

the beginnings of infinitesimal cal-
culus in school instruction 223;
see also curriculum proposals and
reform in instruction.

proposals:

Dresden for training teachers 2.

Regular bodies, groups of 120124.
Rieman surfaces 105 110.

sphere 105110.
Rotation of space 73-

and expansion of space 67 73.

School instruction:

treatment of fractions 27.

rrational numbers 37.

complex numbers 75.

the pendulum 187-190.
exposition of the formal rules of

operation 10.

introduction of negative numbers 22, 28.
notion of function 205-
infinitesimal calculus 221 et seq.
exponent and logarithm 144146,

155-156.
operations with natural numbers

6-8.
trigonometric solution of cubic

equation 134137-
transition to operations with letters 8.
uniformization of the pure equa-
tion by means of the logarithm

133-134.
number-theoretic considerations

37-38.

mathematics, contents of 4

274

Index of Contents.

Signs, rule of 24.
quasi proof for 26.
Space perception 35.
Square root expressions:

significance of for constructions with
ruler and compasses 50.

classification of 53-
Sturm's theorem, geometrical equivalent

of 94.
Style of mathematical presentation 84.

Taylor's formula 223, 233-

analogy with Newton's interpolation
formula 232 et seq.

remainder term 226, 231.
Teachers, academic education of 1.
, academic and normal school training

of 7.

Tensor 63, 66, 70, 72.
Terminology, different in the schools :

algebraic numbers 23.

arithmetic 3.

relative numbers 23.
, misleading in:

algebraically soluble 140.

irreducible 136.

root 140.

Maclaurin's series 224.
Threshold of perception 35.
Transcendence of e 237 243.
- of n 243-249.

Triangle, notion of in spherical tri-
gonometry:

elementary 175.

proper and improper 181 182.

with Mdbius 176, 177, 182-183-

with Study l8l.

triangular membranes 183 186.
Trigonometry, spherical 175 186.

its place in geometry of hyperspace
178-182. .

supplementary relations of 1 83 1 86.
Trigonometric functions, see circular

functions.
Trigonometric functions:

calculation of 170 174

definition by means of quadrature
of circle 162 et seq.

complex fundamental function for
165 et seq.

real fundamental function for 166
et seq.

function - theoretic discussion of
167169.

application of to spherical trigono-
metry 175 186.

application of to oscillations of
pendulum 186190.

application of to representation of
periodic functions 190200; see
also trigonometric series.

series 190200.
Gibb's phenomenon 199-
approximating curves 194196.
convergence, proof of 196198.
trigonometric interpolation 190-193.
behavior at discontinuities 197 et seq.

Uniformization 133, 138.

by means of logarithm 134, 159-

Vector 60, 63-65-
Versor 72.

Internet Archive Audio

Featured

Top

Images

Featured

Top

Software

Featured

Top

Books

Featured

Top

Video

Featured

Top

Mobile Apps

Browser Extensions

Archive-It Subscription

Save Page Now

Full text of "Elementary Mathematics"

See other formats